Labyrinth

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Pilgrimage

Well-placed in the nave (on the threshold of the entrance to the Cathedral of Fulbert, which existed before it), well preserved and truly beautiful, it is one of the largest labyrinths among all existing cathedrals.
This path unfolds over an area of 261.55 m2 (2815 sq feet) and separates the cathedral in 3 and 4 arches under the beams.
This labyrinth is a rare medieval heritage. Others existed in the cathedrals of Reims, Sens, Arras, Auxerra. Today there are still labyrinths in Amiens, Saint Quentin and Bayeux, in the chapter hall (where the assemblies are held). These labyrinths, unlike the one in Knossos, are not to mislead those who walk through them! There are no false routes, impasses, the intricacies of this immense “hopeless” always lead to the center of the drawing.

The labyrinth is a symbolic path that leads man from earth to God, a path where man goes to reencounter God. The center of this great figure symbolizes the City of God. The route through the labyrinth is not just about going to the center, but about sharing. The pilgrim is invited to figuratively assume the line drawn before him towards the heart of the cathedral, towards the east, towards the light.

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The 13th century pilgrim walked this path begging, imploring, as if he or she were making a pilgrimage to Jerusalem, which is why the labyrinths are also called the “Way of Jerusalem”.

After all, what is the Labyrinth?

There are two types of labyrinths mazes, which in English are called:

1. Labyrinth
2. Maze

Labyrinth is what the French text which was the source and refers to: Defined path, with entrance and exit.

Maze is one such as the Minotaur of Greek Mythology. Labyrinth is this one from Chartres.

While both the maze and the “Labyrinth” describe a complex and confusing series of paths, the two are different. A maze is a complex (multicursal) branching puzzle that includes path and direction choices, whereas a labyrinth is unicursal, meaning it has a single, unbranched path that leads to the center of it.

	Labyrinth vs Maze
Meaning	A labyrinth has a single through-route with twists and turns but without branches.	A maze is a confusing pathway that has many branches, choices of path and dead-ends.
Level of difficulty	A labyrinth is not designed to be difficult to navigate. It may be long but there is only one path (unicursal).	A maze is a tour puzzle and can be designed with various levels of difficulty and complexity.
Entry and exit	A labyrinth has only one entrance and that is also the exit. There is just one path from the entrance to the center.	A maze may have different entry and exit points.
Significance	Some labyrinths have a spiritual significance. They signify the complex and long path to reach God.	Mazes are used in science experiments to study spatial awareness and (sometimes) intelligence.

Difficulty

A labyrinth may be complex but is not difficult to navigate because it has a single unambiguous route to the center and back. Mazes can be constructed with varying levels of difficulty and complexity.

Quest for God – Spiritual Meaning – Brief Analysis

The idea of ​​a labyrinth, both in the case of which has an exit and the one which has no exit, or is difficult to exit, suggests something where one turns inside. That is, a wheel. Amazingly, of all the subjects dealt with here, the one that generates the most mismatched, even incoherent information is the labyrinth.
In general, when researching the subject, automatic formulas appear, which teach the manufacture of labyrinths in a variety of ways, from gardens, masonry constructions, rugs, etc. or the idea is related to a variety of things such that it would allow, for example, a complete understanding of the Medieval Age.
In my opinion, for our case, the best site on the subject is Dr. Dan Johnston, Ph.D, a psychologist, affiliated with Mercer Health Systems, in Macon, Georgia. He is Assistant Professor of Psychiatry and Behavioral Sciences at Mercer University School of Medicine in Macon, Georgia. He is the author of the book “Lessons for Living: Simple Solutions for Life’s Problems” (ISBN 0-9712165-0-9) by Dagali Press and creator of the Awakenings Web Site awakenings .
Although I have some restricitons, as this site feels that it is as if the “sauce was more important than the fish”, which is observed in many cases similar to this one. There is a lack of proportionality as to the role or fitting of the labyrinth within a process such as our case. Which can be individuation, as would be the case with Jung’s system, or the completion or integration of opposites, as Jung also foresees and I believe is the ultimate goal of those who approach all of this.
I will synthesize what is there.
First of all, if we examine Dr. Johnston’s entire website, it’s clear that he focuses his objective on the moment a person finds itself in a “problem state”.
The “problem state”, is defined by Jung as one in which the quiet and undisturbed functioning that should characterize a healthy mind is shaken by a shift to the background of this peaceful functioning, with the front-line or main scene taking in the mind of a problem that worries or troubles the person.
This, of course, can be an illness, from schizophrenia, to an obsession or even a psychosis. In reality, it is the intensity and the way in which the individual treats his ideas that classify him according to psychiatric or psychological criteria.
Let’s exclude the unhealthy states and focus on the states that occur for the person to grow up.
These “trouble states” can have varying degrees of intensity and depth.
To a lesser degree, it may just be a passage in which the person is learning to deal with something that, without being minor, has to have a clear definition. For example, finance. A person can get involved in complications because he or she doesn’t know how to handle money and has to learn how to do it. It can be anything from a simple trip to the bank or to bankruptcy. It generates a state of the type we are discussing here and which I classify as “minor”.
Another example, also “minor”, ​​is the case of a person who is on the rise professionally, is promoted to a managerial position but does not know how to command. He or she is harassed or disrespected by some subordinate. The inexperience can put he or she in a state of trouble.
In short, life practically only presents situations of this type to a greater or lesser degree.
In a “greater” degree, par excellence, are romantic involvements, marriage, separation, midlife crises, heart attacks, serious illnesses, death of very close people, threat or proximity to death.
It is obvious that visits to the psychologist or psychiatrist, when not motivated by unhealthy states, always have the above situations in their context.
All forms of treatment used by psychology or psychiatry aim to overcome, remedy or at least live and accept these problems. The goal is to recover what should be the normal atitude towards life, when the problem fades into the background and the smooth running of a balanced mind comes back to inhabit one’s mind.
Although this site makes no distinction between these situations, it only mentions, we can use its content to understand the relationship between the labyrinth and them and perhaps shed some light on why humanity has the labyrinth as an archetype. It covers the following aspects:

1. Choosing Attitude: Do you want to be bitter or better?
2. The Change Cycle and the predictable process that follows.
3. The Enneagram: Found the “Inner Self”
4. The labyrinth – walking a spiritual path
5. The labyrinth hidden in the stereogram.
6. Life lessons.
7. Midlife metamorphosis: how to be brought to completion.
8. Mindstorms. Don’t get carried away.
9. Understanding depression.
10. Teaching through stereograms.
11. How to deal with stress.
12. Wheel of life 1: feelings in change.
13. Wheel of Life 2: The Spiritual Journey to Your Center

1. Choosing Attitude: Do you want to be bitter or better?

We cannot escape the fact that we are responsible for everything that happens to us in life (not in life). What’s the difference? What comes from life we cannot avoid, what we do in life is our problem. For example, we get a job in life, but we can lose it in life…
You are not responsible for everything that happens to you, but you are responsible for how you react. Life acts. You react.
This is attitude. It’s made of:

• What do you think
• What do you do
• What do you feel

The first two you have control, the third one you don’t. Unfortunately, that’s what counts most. We don’t know in our conscious thought what or why we feel. The work is “knowing yourself” as the Greeks said thousands of years ago.
We are what we think we are, but we are not what we think we are.
Our rules, our way of being, what we are, if we want to know, is one of the most arduous tasks a human being can engage in.

2-The Change Cycle and the predictable process that follows.

A change in life can be understood as a 6-step repetitive process, with two central characteristics which are strength and resilience, which is the characteristic of elasticity or rapid recovery.

1. Something´s up

The first step is early recognition of a change that is about to take place. Everything looks normal, but you get a feeling that something is different. It’s hard to pinpoint precisely, but you feel more than you know something is going to happen

2. What is it?

It is the step of clarifying and recognizing the situation. It could be a problem or an opportunity. It has to be clearly defined which of the two it is, as it can represent a danger or a chance for growth.

The Chinese crisis ideogram is composed of two separate characters: one means danger and the other opportunity. One translation is that a crisis is a dangerous opportunity. The elements in the crisis of danger and opportunity must be carefully recognized. More than in any other situation, it is necessary for us to get to know who we are to avoid danger and seize opportunities

3.What Do I Need To Do?

Once the situation is recognized, you need an action plan. What do you have to do and how are you going to do it? Technically it is more of an implementation plan, which can be formal or informal and has to be developed.

4.I Will Do This!

Once plans are made, they have to be put into action. You, knowing what has to be done, you do it.

5.I Am Able To Do This.

As the action plan is used and modified and improved, it reaches a point where it feels like it’s working. There is a feeling that the goal can be achieved.

6.I Got Through It. What’s Next?

Celebration time! The change was made, things go back to normal. Not the “old normal”, but the “new normal”. We also know that changes are taking place and we look forward to what awaits us.
As I said earlier, the author discusses what I call the “minor problem”. Applying this recipe, for example, to breaking up a marriage just doesn’t work. The concept serves to situate the process, the steps and the solution are more complex.
I believe there are labyrinths and “maze” for this. What the author describes above is a kind of “maze”, that is, the exit exists but has to be found.
In the case of the labyrinth, that the path is ready, in my view it represents the answer to a problem that has the characteristics of a paradox and needs the union of opposites. There has to be something ready, complete. How come? If the “maze” when you are experiencing a conflict of opposites has no way out, the process has to be to overcome the situation and arrive at a solution that integrates everything and appears clear and complete.
When a person arrives in Chartres, or in Compostela, or in Aparecida do Norte (even without a labyrinth), he or she is at a point in the process that is finalizing and enshrining something from the interior that was brought into balance and resolved or will be resolved there.
Jung is very strong in that. He discusses the problem of paradox (or opposites) and categorically states that we never solve problems, we just get past through them.
It’s a shame that a site as well-designed as Dr. Johnston’s doesn’t discuss this extensively.
I am convinced that for the problems I classify as minors, psychologists and psychiatrists are enough and for the problems I call major, the bearer has to discover the meaning of life.
In my view, the meaning of life cannot be found without the idea of ​​religion. In fact, the very meaning of reconnecting is saying this.
Unfortunately, or fortunately, the cleric, the minister, or the rabbi is not always the counterpart of the psychologist and the psychiatrist.
For those who are dedicated to science or to some cause, such as communism, as a meaning of life, it is good to be aware because they do it in the same way as a religious person. The difference is that it puts faith in sets of ideas that, however much they make sense, are not entirely true and has proven to not work.
The culture has taken us to the point that in many cases this trip is lonely. In fact, this situation is not new, because in Homer’s Odyssey, the saga of the hero Odysseus, or Ulysses, as he is better known, has a circumstance in which he sails attached to the ship’s mast, with everyone else with their ears covered. Only he hears the sirens’ song and as it was dangerous and generated the possibility of madness or deception, he was tied to the mast.
Nothing more perfect to illustrate the problem state of those who are looking for the meaning of life. It is a perfect metaphor for a Labyrinth with no way out or a difficult way out.
Jung, in a rather dense and complex form, as is characteristic of him, analyzes this in his text: Psychotherapists or the Clergy

3.The Enneagram: Finding the “Inner Self”

We all have a hidden part of our self that is the reality of who we truly are. The classic work that had endless filmed versions of Dr Jekyll and Mr. Hyde. Perhaps the most widespread (and best accepted) concept of this fact is that of Freud. It’s the famous Id.
In the case of Freud, our Mr. Hyde is motivated by the libido and psychoanalysis aims to make us see this and “domesticize” or at least live with it.
Jung extends this idea and in his perception, our Mr. Hyde yearns for meaning in life and re-encounter with divinity.
We can use drugs or behavioral therapies to solve “problem states”, but unless the circumstances call for it or justify it, it is the same as covering the sun with a sieve.
There are a multitude of ways to represent this characteristic of our way of being, many including denying the existence of it or even of having a thought driving our action.
It is not possible to overcome a “problem state” without some kind of action that neutralizes it, recognizes it, manages it, lives with it, or etc. It is not the place to discuss the best way or the best concept about this fact. Dr. Johnston opts for the enneagram as a self-knowledge tool.
Enneagram is not a horoscope, but it is to similar to it. Perhaps an I Ching a little more clarified.
Although Jung does not cite the Enneagram in any of his works, Jung’s typology is surprisingly consistent with that of the Enneagram.
There are serious business consultancies in the United States using eneagaram to solve problems of team matching and people motivation in large companies.
For those familiar with Jung’s typology, the enneagram looks familiar.
I have no position on the enneagram. I think it’s a good start. When it comes to self-knowledge, “you get what you pay”, that is, you get what you pay for. The more magic and the less effort, the less insight, and the more reality and effort, the more insight.

4.The labyrinth – walking through a spiritual path

“Your life is a sacred journey. And it’s about change, growth, discovery, movement, transformation, continually expanding your vision of what is possible, expanding your soul, learning to see clearly and deeply, listening to your intuition, courageously taking on challenges every time. You are on the path… right where you should be now… and from here, you can just go forward, shaping your life like a magnificent story of triumph, of healing courage, of beauty, wisdom, power, dignity, and love.”

Caroline Adams

At this point I justify my opinion that “sauce is more important than the fish it is cooked with”. If we click and go to the site, we will verify on the one hand a careful research on labyrinths. That is worth examining. On the other hand, there is an exaggerated appreciation of form, as if, suddenly, in 15 minutes, walking on the floor of Chartres Cathedral, everything that is described there by Caroline Adams materializes. No way! “You get what you pay”.

5.Stereogram

It is a practical way to create an initiatic spirit without esoterism. I am nearsighted and have eyestrain and have difficulty with visual experiences of this nature. But it’s still interesting.

6.Lessons for living

Encouragement messages

7.Midlife crisis

Dr. Johnston carefully does not use the jargon “mid life crisis” using instead Midlife Metamorphosis. I read the quoted author Gail Sheehy, and I find the text weak. I think the weakest part of this site is this one.
For mid life crisis Jung is unbeatable. The cover of the “Portable Jung”, which is a collection of Jung’s works that Campbell made, is a representation of the mid-life crisis. Elsewhere, I present an appreciation of this work within a text I call a link. See the part where Jung discusses it: Modern Man in Search of a Soul C G Jung , THE STAGES OF LIFE. pages 109-132

8.”Mind Storms”

He minimizes what I described above as “problem state” and over simplifies it.

9.Depression

Depression, which tends to accompany “problem states” of the “major” type..

10.Sidewalks of life

He compares life to a walk along the sidewalk and proposes a poem that resembles Drummond’s poem about the stone blocking the way or the music: March Waters

A stone blocking your way

Carlos Drumond De Andrade

In the middle of the way there was a stone
there was a stone in the middle of the way
there it was a stone
in the middle of the way there was a stone.
I will never forget this event
in the life of my so tired retinas.
I’ll never forget that halfway
there it was a stone
there was a stone in the middle of the way
in the middle of the way there was a stone

Os Companheiros

A igreja catedral de Chartres foi construída por trabalhadores especializados.
Os trabalhadores do gótico, construtores de igrejas. Os “Operários”. Eles deixaram, sobre as pedras que talharam, sobre as vigas que montaram, sinais gravados que são suas marcas, sua assinatura. Afora isto, não sabemos mais nada deles. Sua origem é misteriosa e se tornou lendária.

Quando criamos uma lenda, é para transmitir qualquer coisa para uso daqueles que não tem a chave. De qualquer maneira, perde-se a chave e a historia com ela. A lenda, somente, fica. Separar a historia da lenda da margem a inúmeras possibilidades de erro. A lenda é às vezes muito clara e as vezes menos clara, quando devemos recorrer à hipótese. Dizemos que os construtores de igrejas se reuniam em confrarias, o que seria, na verdade, mais correto chamar de “fraternidades” ou” associação de operários”.

Existiam três fraternidades: os Infantes do padre Soubise, os Infantes do Mestre Jacques e os Infantes de Salomão. Estas fraternidades não desapareceram completamente. Os Infantes deixaram herdeiros que são conhecidos, nos dias atuais, pelo nome de Companheiros dos Deveres da Torre da Franca, nome que lhes foi dado no século XIX.

Algumas destas fraternidades parecem ter conservado uma tradição iniciatica, outras não. Mas todas guardam uma tradição “do metier”, uma tradição moral de cavalheirismo e de submissão à obra que deve ser conseguida. Uma historia, um conto, resume seu objetivo:

Três homens trabalham num canteiro de obras. Uma pessoa que passa lhes pergunta:

– Que fazem vocês?

– Eu ganho meu pão, diz o primeiro.
– Eu faço meu trabalho diz o segundo.
– Eu faço uma catedral, diz o terceiro.

Este ultimo é um companheiro.

A palavra companheiros, (compagnons no original Francês) etmologicamente quer dizer indivíduos que repartem o mesmo pão. Mas esta etimologia não estão sozinha. Para Raoul Vergez, os companheiros são aqueles que sabem utilizar o compasso.

Os indivíduos que compartilham o mesmo pão formam uma comunidade, uma fraternidade, os indivíduos que sabem utilizar o compasso são pessoas que tem certos conhecimentos de leis geométricas de harmonia que lhes permitem ascender ao estagio de “trabalhador especializado.”

Perseguidos na época do processo contra os Templários por Felipe, o Belo, interditados pelas corporações, eles assumiram o nome de Companheiros dos Deveres e entraram na clandestinidade somente para sair à época da Revolução Francesa, que destruiu as corporações.

Eles se reconhecem entre si pelas palavras, pelos sinais e pela gíria do próprio trabalho e da clandestinidade. O termo dos deveres guarda para eles, todo seu sentido: dever de operário, que foi para eles, outrora, o meio de conseguir seus objetivos, dever profissional e humano que não sejam jamais desmentidos.

Suas tradições permanecem vivas e se experimenta, sob o plano “atividade” por meio de uma hierarquia de três graus: Aprendiz, Companheiro e Companheiro-acabado ou Mestre. No plano humano, promotores de uma obra vital, eles se recusam a portar armas e se forem formalmente solicitados, Cavaleiros que são, isto é, libertadores, eles sempre se negam a construir fortalezas ou prisões. Eu creio que recusam até hoje.

Eu não creio os estar traindo dizendo que seu pensamento mais profundo é que o homem vale aquilo que é capaz de fazer. Isto não agradaria de forma alguma os sindicatos modernos!.

Os aprendizes aprendem seu trabalho, de canteiro em canteiro de obras num curso de um Giro pela França, sob a direção dos companheiros ou de qualquer outro, mas o saber particular de suas confrarias lhes é ensinado separadamente, nos aquartelamentos, pelos mestres.

As três Fraternidades que outrora lutaram entre si, estão nos dias de hoje reunidas em uma só associação, mas parece bem que a origem dos seus “deveres” e de suas técnicas foram diferentes. .

Os Infantes do Padre Soubise tem a fama de terem sido criados por um monge beneditino legendário (Alem do fato que perto de Poitiers um bosque de um monastério beneditino tem o nome de Bosque do Padre Soubise ). Foi ele que ensinou os companheiros.

É provável que se trate de uma fraternidade criada entre os beneditinos mesmo, ou então de “laicos” desta atividade que desfrutavam da proteção, necessária nesta época, das casas conventuais motivo pelo qual eles usavam – ou não usavam – a aparência de monges.

E porque o romano é beneditino, eu tenho tendência a crer que foi esta fraternidade dos Infantes do Padre Soubise que construíram, com a ajuda dos monges construtores, abadias, igrejas e catedrais romanas.

Na época das perseguições que sofreram as fraternidades no século XIV eles se recusaram a se separar da Igreja. .sp Uma outra fraternidade de companheiros era a dos Infantes do Mestre Jacques. Eles se transformaram em Companheiros Passantes do Dever. Sua lenda é cheia de poesia.
Seu fundador teria sido Mestre Jacques, que nasceu em uma pequena cidade Gaulesa chamada Carte (Saint-Romilly nos dias de hoje, segundo Lucien Carny, em “Atlantis”, nr.222.) no Sul da Franca.

O pai de Mestre Jacques teria sido o mestre de obras Jacquin, que se tornou mestre depois de suas viagens à Grécia, ao Egito e à Jerusalém, onde ele teria executado duas colunas do Templo de Salomão (Uma efetivamente chama-se Jaquin).

Eles são passantes. Eu tendo a pensar que este particípio verbal não é de um verbo reflexivo, mas que designa pessoas que “fazem passar”.

E quando sabemos dos obstáculos que representaram, ha muito tempo atras, os rios aos viajantes, imaginamos que atravessa-los, quer seja em pontes suspensas ou empontes mesmo, constituía uma seria cooperação à obra da civilização.

Pode ser que eles tenham sido os herdeiros dos Monges construtores de Pontes que foram grandes construtores.

A menos que sua lenda lhes seja anterior, pois que tem raízes longínquas, como o nome mesmo de Jacques, que ha muito tempo atras designava o camponês gaulês. Seria admissível que os Infantes do Mestre Jacques tenham sido herdeiros desta confraria de construtores celtas que assinavam com a figura de uma folha do carvalho.

Enfim, podemos pensar que eles tenham se constituído em fraternidade para cumprir a tarefa de organizar o equipamento religioso e hoteleiro sobre o caminho de Saint Jacques da Compostela.

E a terceira confraria, os Infantes de Salomão a quem eu atribuirei voluntários para a construção não apenas de Chartres, como também de uma boa parte das Notre-Dames góticas e, de qualquer maneira, embora elas não aparecem “assinadas”, eu também atribuiria a construção das catedrais de Reims e Amiens.

E eis porque as razões:.

Os Infantes do Mestre Jacques pareciam estacionados, pelo menos na época de sua passagem para a clandestinidade, na Aquitania. Suas igrejas, ornadas da crista da espada ou da cruz com aparência celtica, contornada de um circulo, não são encontráveis, salvo raras exceções, senão no sul da Franca. Elas tem, alias, um estilo muito pessoal.

Os Infantes do Padre Soubise, beneditinos, pareciam mais voltados ao estilo romano e as marcas de sua companhia de construtores do romano diferem grandemente da dos construtores do gótico, mesmo em se tratando de monumentos contemporâneos.

E como deveria ser, necessariamente, uma fraternidade de construtores góticos, não se poderia tratar senão dos Infantes de Salomão. Alem disto, na casa dos seus herdeiros, que se transformaram nos Companheiros do Dever da Liberdade, que permaneceu a tradição do ensino da geometria descritiva necessária, do “Traçado”, pelos monges de Citeaux.

Eles teriam sido, neste caso, uma fraternidade de construtores religiosos, criados pela Citeaux, paralelamente, se formos ver, a Ordem do Templo e sua “proteção”, até a aquisição de privilégios teria sido confiada a eles.

O nome mesmo de Salomão poderia bem ser uma indicação suplementar. Salomão, o Adepto, fez construir o Templo e ai enterrou a Arca.

São Bernardo, cisterciano, acreditou ter desenvolvido, em cento e vinte sermões, aos seus monges, o livro do Adepto de Salomão: O cântico dos Cânticos. São Bernardo criou a Ordem do Templo cujo nome primitivo foi Templum Salomonis: do Templo de Salomão. Os cistercianos ensinavam os Infantes de Salomão. .

Juntemos ainda a isto o fato da demanda de Amaury, prior dos Gauleses para o Templo, a quem estava ligado por amizade S.Luis, que conciliou as fraternidades laicas de construtores de igrejas com privilégios que dispensavam a necessidade de ter que recorrer à proteção de estrangeiros às suas fraternidades. .

Uma questão ainda ficaria pendente: Que condições de coexistência haviam com a Ordem do Templo? Sua posição era de estarem dentro da Ordem, somente afiliados ou ainda eram eles apenas associados? .

É difícil responder, pois o Templo tinha uma organização muito complexa, mistura de monges e laicos, de militares e artesãos, todos designados sob o nome de “irmãos”.

Na sua própria base, estavam os Frades de convento, que, homens de armas ou não, eram monges, pois tinham pronunciado seus votos. Estes eram os verdadeiros Templários.

Na organização geral existiam, alem disto, voluntários, servindo pela vida toda ou por algum tempo, dentre os quais estavam os Frades do metier que podiam, muito bem, ser os construtores reunidos com os Infantes de Salomão. .

Vê-se como é difícil responder.

Uma indicação, todavia: os Cavaleiros, nas comanderias, habitavam um aposento interditado aonde não podiam entrar as mulheres e apenas os “convidados”. Esta era a “Grande Casa”, o convento propriamente dito.

A “Grande Casa” é, evidentemente, assim chamada em oposição à “Pequena Casa”. Ou em gaulês, nome conservado nos países Normandos, e sem duvida também na Picardia, a pequena, o aquartelamento. .
É o aquartelamento é, tradicionalmente, um lugar reservado à Fraternidade de construtores.

Podemos portanto admitir logicamente que perto da Ordem maior dos Cavaleiros do Templo de Salomão, existia uma ordem menor e , de alguma maneira filiada à Ordem maior: os Infantes de Salomão. .
De qualquer maneira, ao mesmo tempo que executava o processo do Templo, Felipe, o Belo, suprimia os privilégios dados aos “maçons”. ( Pedreiros, em Francês).

Se os Infantes do Padre Soubise se inclinaram, não ocorreu o mesmo com os Infantes de Salomão que, depois de haver criado algumas confusões, passaram à clandestinidade ou, para muitos, se expatriaram.

Se considerarmos como desligados de toda obrigação para com o rei da Franca o inverso com relação ao Papado, culpado de não ter defendido a Ordem, eles se tornaram os Companheiros estrangeiros do Dever de Salomão. .

Será preciso dizer que existem certas coincidências entre as “assinaturas” da Fraternidade que construiu as Notre Dame se certas construções templarias?

O pilar acantonado

A inversão da cruz celtica, base do pilar acantonado.

As duas soluções de Chartres para o pilar acantonado.

A Fraternidade, especialmente em Chartres, Amiens e Reims, pareciam ter assinado estes monumentos pelo emprego do pilar cantonado.

Sem duvida o aspecto não é quanto à sua invenção, pois nós os encontramos bastante nas construções romanas, mas é quanto à sua aparência particular, no que toca às suas proporções, que os personaliza de uma certa maneira.

O pilar é redondo e octogonal em Chartres mas as colunatas “acantonadas”, isto é, dispostas em cruz, apresentam a particularidade de ter, em relação ao pilar central, a mesma proporção que os “pequenos” círculos que penetram em corte o circulo central da cruz celtica em relação ao circulo.

Em Amiens, Reims e Chartres, reencontremos esta “assinatura”, que poderia ser de uma “escola”, nos dois pilares da Notre Dame de Paris, perto da porta Ocidental, e duas em Beauvais, uma em cada transepto.

Sem querer tirar conclusões prematuras, somos, malgrado o que seja, tentados a comparar este pilar com o clássico torreão templário, a torre acantonada de quatro torrezinhas, tal e qual podemos ver em Sarzay, no Indre, ou mesmo no Torreão de Vincennes, copia da Torre do Templo de Paris.

Existe, igualmente, algum parentesco com a Torre de Cesar, em Provins, uma obra provável dos construtores da fortaleza do Templo. A relação com os pilares octogonais das colunas redondas “saltam aos olhos”.

Outros indícios ainda: Em Reims, Amiens e Chartres,no Portal Real, nos cordões da superfície da abobada, estão representados dois cavaleiros – algumas vezes despidos, como em Reims – protegidos por um só escudo contra uma granada.

Lembra a dualidade templaria?.

E a granada é um símbolo alquimico.

Outras coincidências históricas se apresentam: O gótico verdadeiro, nascido ao mesmo tempo que o Templo, termina com ele. Ele se transforma em “formato ogival” ou “gótico florido ou flamejante”. Não é mais que uma virtuosidade. Não é mais o Templo iniciatico. .

Da mesma maneira, o vitral desaparece. Ele cede seu lugar ao vidro pintado que não vale mais do que a pintura e, em todo caso, não tem outra virtude que as do seus adornos.

É necessário, para concluir, dizer que os “conhecedores”, os “sábios”, se encontravam no Templo e desapareceram com ele?

Continua em: O Tesouro do Templo

Réalité et chiffres

                            

Veja em Português        See it in English

L’ensemble de ce site/blog reposait sur des études, des visites et des observations faites entre 1983 et 2000. Le premier format similaire à ce qui est devenu la norme sur Internet s’est produit entre 2000 et 2010. La contradiction que je ressens entre la réalité que je perçois et l’utilisation des chiffres pour le décrire et qui est décrit dans l’article original, je ne savais pas qu’il faisait l’objet d’une étude formelle et j’ai fini par le découvrir par accident maintenant, en 2020, afin d’écrire des articles de chronique sur la science que j’ai dans le journal local de ma ville.
Formellement, à partir de la recherche que j’ai faite, des études significatives soulignant cette divergence, sont dans la discussion “Quelles preuves seraient nécessaires pour prouver que la nature est intrinsèquement mathématique?”
J’ai indiqué cette question posée sur la plateforme Research Gate, pour mémoire, car je pense que cela ne résout pas le problème, cela ne fait que le compliquer et, franchement, cela devient ennuyeux et fastidieux. J’essaierai de résumer sous une forme plus compacte et intelligible pour autant que je puisse comprendre.

Qui a dit que l’univers était composé de nombres? Galilée, alias qui a «inventé» ce qui allait être connu sous le nom de Science, en mesurant l’Enfer de Dante.
L’idée que tout est en quelque sorte mathématique remonte au moins aux Pythagoriciens de la Grèce antique et a engendré des siècles de discussions entre physiciens et philosophes. Au 17ème siècle, Galilée a déclaré que notre univers est un «grand livre» écrit dans le langage des mathématiques.
L’efficacité irrationnelle des mathématiques en sciences naturelles est le titre d’un article publié en 1960 par le physicien Eugene Wigner. Dans ce document, Wigner a noté que la structure mathématique d’une théorie physique ouvre souvent la voie à des avancées futures de cette théorie ou même à des prédictions empiriques.
Il ouvre l’article avec une citation de Bertrand Russel:

Cette prémisse se heurte à la suivante

«Les mathématiques, correctement vues, ont non seulement la vérité, mais la beauté suprême – une beauté froide et austère, comme celle de la sculpture, sans appel à aucune partie de notre nature la plus faible, sans les beaux ornements de la peinture ou de la musique, bien que sublimement pur et capable d’une perfection sévère, comme seul le meilleur art peut le montrer. Le véritable esprit de plaisir, d’exaltation, le sentiment d’être plus que l’homme, qui est la pierre de touche de la plus haute excellence, se retrouvent à la fois dans les mathématiques et dans la poésie.

Correctement visualisé

Cette prémisse se heurte à ce qui suit:

Apparences et choses en elles-mêmes

Dans la première édition (A) de la Critique de la raison pure, publiée en 1781, Kant plaide pour un ensemble surprenant de revendications sur l’espace, le temps et les objets:

• L’espace et le temps ne sont que les formes de notre intuition sensible des objets. Ce ne sont pas des êtres qui existent indépendamment de notre intuition (les choses en elles-mêmes), ni des propriétés ni des relations entre de tels êtres. (A26, A33)
• Les objets dont nous avons l’intuition dans l’espace et le temps sont des apparences, non des objets qui existent indépendamment de notre intuition (les choses en elles-mêmes). Cela est également vrai des états mentaux que nous intuitons dans l’introspection; dans le «sens intérieur» (conscience introspective de mes états intérieurs), j’intuit seulement comment je m’apparais, pas comment je suis «en moi». (A37–8, A42)
• Nous ne pouvons connaître que des objets que nous pouvons, en principe, intuiter. Par conséquent, nous ne pouvons connaître que les objets dans l’espace et le temps, les apparences. Nous ne pouvons pas connaître les choses en elles-mêmes. (A239)
• Néanmoins, nous pouvons penser aux choses en elles-mêmes en utilisant les catégories (A254).
• Les choses en elles-mêmes nous affectent, activant notre faculté sensible (A190, A387). ^[1]

Raymond Tallis on maths’ unreasonable effectiveness (excerpts) 

Raymond Tallis sur l’efficacité déraisonnable des mathématiques (extraits)

La plus grande preuve que la réalité n’est pas mathématique est peut-être que le monde des lois physiques – qui permet des prévisions de quantités – est un monde de quantités sans qualités.
Ce n’est pas un oubli accidentel. Galilée, qui a lancé la révolution scientifique, a fait valoir que les couleurs, les goûts, les sons, les odeurs n’avaient pas leur place dans le monde matériel, dont le livre était écrit en mathématiques. Les qualités que nous expérimentons ont été introduites par les êtres sensibles. Par contre, la réalité physique elle-même était composée de «qualités primaires» telles que la taille, la forme, l’emplacement et le mouvement, qui peuvent être exprimées en termes mathématiques sans laisser de résidu. Ainsi, l’univers mathématique de la physique n’a pas ce qu’il est aujourd’hui (d’après John Locke) généralement appelé par le terme légèrement péjoratif de «qualités secondaires». Ceux-ci résistent à être mathématisés. Les mathématiques de la lumière ne se rapprochent pas de l’expérience du jaune, et la description mathématique des schémas d’influx nerveux n’affecte pas la douleur elle-même. Cela est parfois considéré comme une preuve que ni la couleur ni la douleur ne sont vraiment réelles – bien qu’il puisse être difficile de vendre cette déclaration à l’homme qui regarde un narcisse ou une femme ayant mal aux dents.

Et il y a autre chose qui manque: l’actualité. Les lois physiques décrivent les relations plus générales entre les modèles de changement. Mais un modèle de relations entre événements ou objets n’est pas un événement ou un objet singulier, réalisé à travers les qualités qui les caractérisent. Wigner a souligné quelque chose à cet égard lorsqu’il a observé que «les lois de la nature sont toutes des déclarations conditionnelles et ne concernent qu’une très petite partie de notre connaissance du monde … [elles ne donnent pas] d’informations sur l’existence, les positions actuelles ou les vitesses de ces corps. »- bref, sur les réalités, toutes en dehors des lois elles-mêmes, étant les conditions dans lesquelles les lois fonctionnent.

Le monde des lois physiques – qui permet des prévisions de quantités – est un monde de quantités sans qualités.

Même ainsi, nous devons encore expliquer l’efficacité «irrationnelle» de la physique et des mathématiques qui sont au cœur de celle-ci. De toute évidence, la physique doit toucher quelque chose de très fondamental. Rejeter les Pythagoriciens contemporains en soutenant que la physique mathématique ne comprend que les aspects quantitatifs de la réalité – d’où leurs prédictions quantitatives extraordinairement précises – laisse inexplicable le fait que ces prédictions permettent également une technologie qui peut façonner le monde – expériences réelles, événements, objets et plus choses – selon nos souhaits. Les mathématiques font plus qu’un simple contact avec nos vies. D’un autre côté, nous ne pouvons ignorer les autres types de vérités, enracinées dans l’expérience réelle des êtres humains qui sont au-delà des mathématiques: des vérités situationnelles saturées de qualités, de sentiments et de préoccupations, et de différenciations d’espace et de temps (“ici”, “maintenant”).

Reality and numbers

                               

Veja em Português        Voir en français

This entire blog site was based on studies, visits and observations made between 1983 and 2000. The first format similar to what became standard on the Internet occurred between 2000 and 2010. The discrepancy I feel between the reality I perceive and the use of numbers to describe it and which is described in the original article, I didn’t know it is the object of formal studies and I ended up discovering it by accident now, in 2020, in order to write column articles on science that I have in the local newspaper of my city.
Formally, from the research I have done, the significant studies pointing out this discrepancy, is summarized in the discussion “What evidence would be needed to prove that nature is intrinsically mathematical?”
I have indicated this question made on the Research Gate platform, for the record, because I think it does not resolve the issue, it only complicates it and, frankly, it gets boring and tedious. I will try to summarize in a more compact and intelligible form as far as I could understand.

Who said the universe is made up of numbers? Galileo, alias who “invented” what came to be known as Science, measuring Dante’s Inferno.
The idea that everything is, in a way, mathematical, dates back at least to the Pythagoreans of ancient Greece and has generated centuries of discussions between physicists and philosophers. In the 17th century, Galileo stated that our universe is a “great book” written in the language of mathematics.
The Irrational Effectiveness of Mathematics in Natural Sciences is the title of an article published in 1960 by physicist Eugene Wigner. In it, Wigner noted that the mathematical structure of a physical theory often points the way for future advances in that theory or even empirical predictions.
He opens the article with a quote from Bertrand Russel:

“Mathematics, correctly seen, has not only the truth, but the supreme beauty – a cold and austere beauty, like that of sculpture, without appeal to any part of our weakest nature, without the beautiful ornaments of painting or music, although sublimely pure and capable of severe perfection, as only the best art can ”show. The true spirit of delight, exaltation, the feeling of being more than man, who is the touchstone of the highest excellence, can be found in both mathematics and poetry.”

Correctly viewed

This premise comes up against the following:

Appearances and Things in Themselves

In the first edition (A) of the Critique of Pure Reason, published in 1781, Kant argues for a surprising set of claims about space, time, and objects:

• Space and time are merely the forms of our sensible intuition of objects. They are not beings that exist independently of our intuition (things in themselves), nor are they properties of, nor relations among, such beings. (A26, A33)
• The objects we intuit in space and time are appearances, not objects that exist independently of our intuition (things in themselves). This is also true of the mental states we intuit in introspection; in “inner sense” (introspective awareness of my inner states) I intuit only how I appear to myself, not how I am “in myself”. (A37–8, A42)
• We can only cognize objects that we can, in principle, intuit. Consequently, we can only cognize objects in space and time, appearances. We cannot cognize things in themselves. (A239)
• Nonetheless, we can think about things in themselves using the categories (A254).
• Things in themselves affect us, activating our sensible faculty (A190, A387).^[1]

Raymond Tallis on maths’ unreasonable effectiveness (excerpts)

Perhaps the greatest evidence that reality is not mathematical is that the world of physical laws – which allows forecasts of quantities – is a world of quantities without qualities.

The world of physical laws – which enables predictions of quantities – is a world of quantities without qualities.

This is not an accidental oversight. Galileo, who kick-started the scientific revolution, argued that colours, tastes, sounds, odours, had no place in the material world, whose book was written in mathematics. The qualities we experience were introduced by sentient beings. By contrast, physical reality itself was comprised of ‘primary qualities’, such as size, shape, location and motion, which can be expressed in mathematical terms without residues. So the mathematized universe of physics lacks what are now (after John Locke) usually called by the slightly derogatory term ‘secondary qualities’. These resist being mathematised. The mathematics of light does not get anywhere near the experience of yellow, nor does the mathematical description of patterns of nerve impulses reach pain itself. This is sometimes seen as evidence that neither the colour nor the pain are really real – although it might be difficult to sell this claim to the man looking at a daffodil or a woman with toothache.

And there is something else missing: actuality. Physical laws describe the most general relations between patterns of change. But a pattern of relations between events or objects is not an event or an object, which are singulars, realised through the qualities that characterise them. Wigner pointed out something along these lines when he remarked that “the laws of nature are all conditional statements and they relate only to a very small part of our knowledge of the world… [they give] no information on the existence, the present positions, or velocities of these bodies” – in short, on actualities, which are all outside of the laws themselves, being the conditions on which the laws operate.

Even so, we still need to explain the ‘unreasonable’ effectiveness of physics and of the mathematics that lies at its heart. Clearly physics must be getting something very fundamental very right. Dismissing the contemporary Pythagoreans by arguing that mathematical physics grasps merely the quantitative aspects of reality – hence its extraordinarily precise quantitative predictions – leaves unexplained the fact that those predictions also enable technology that can shape the world – real experiences, events, objects, and stuff – in accordance with our wishes. Maths makes more than a passing contact with our lives. On the other hand, we cannot ignore the other kinds of truths, rooted in the actual experience of human beings that lie beyond mathematics: situational truths saturated with qualities and feelings and concerns, and differentiations of space and time (‘here’, ‘now’).

Realidade e Números

                            

 Voir en français      See it in English

Todo este site blog foi baseado em estudos, visitas e observações feitas entre 1983 e 2000. O primeiro formato semelhante ao que se tornou padrão na Internet ocorreu entre 2000 e 2010. A discrepância que eu sinto entre a realidade que percebo e o uso de números para descrevê-la e que está descrita no artigo original, eu não sabia que é objeto de estudo formal e acabei descobrindo isso por acidente agora, em 2020, para poder escrever artigos sobre ciência em coluna que tenho no jornal local de minha cidade.

Formalmente, pelas pesquisas que fiz, os estudos significativos apontando esta discrepância, está na discussão “Que evidência seria necessária para provar que a natureza é intrinsecamente matemática?”

Deixei indicado este questionamento feito na plataforma Research Gate, para constar, porque acho que não resolve a questão, apenas a complica e, francamente, fica uma coisa enfadonha e entediante. Vou tentar resumir de forma mais compacta e inteligível pelo que pude entender.

Quem disse que o universo é composto de números? Galileu, alias que “inventou” o que veio a ser conhecido como Ciência, medindo o Inferno de Dante.
A idéia de que tudo é, de certa forma, matemática, remonta ao menos aos pitagóricos da Grécia antiga e gerou séculos de discussões entre físicos e filósofos. No século XVII, Galileu afirmou que nosso universo é um “grande livro” escrito na linguagem da matemática.

A Eficácia Irracional da Matemática nas Ciências Naturais é o título de um artigo publicado em 1960 pelo físico Eugene Wigner. Nele, Wigner observou que a estrutura matemática de uma teoria física frequentemente indica o caminho para avanços futuros para essa teoria ou até previsões empíricas.

Ele abre o artigo com uma citação de Bertrand Russel:

“A matemática, corretamente vista, possui não apenas a verdade, mas a beleza suprema – uma beleza fria e austera, como a da escultura, sem apelo a qualquer parte de nossa natureza mais fraca, sem os belos adornos da pintura ou da música, ainda que sublimemente pura e capaz de severa perfeição, como somente a melhor arte pode” mostrar. O verdadeiro espírito de deleite, a exaltação, a sensação de ser mais do que o homem, que é a pedra de toque da mais alta excelência, podem ser encontrados tanto na matemática quanto na poesia.”

Corretamente vista

Esta premissa, esbarra no seguinte:

Aparências e coisas em si mesmas

Na primeira edição da Crítica da Razão Pura, publicada em 1781, Kant defende um conjunto surpreendente de afirmações sobre espaço, tempo e objetos:

• Espaço e tempo são apenas as formas de nossa intuição sensível dos objetos. Eles não são seres que existem independentemente de nossa intuição (coisas em si), nem são propriedades de, nem relações entre tais seres. (A26, A33)
• Os objetos que intuímos no espaço e no tempo são aparências, não objetos que existem independentemente de nossa intuição (coisas em si mesmas). Isso também se aplica aos estados mentais que intuímos na introspecção; no “sentido interior” (consciência introspectiva dos meus estados internos), intuto apenas como me pareço, não como estou “em mim mesmo”. (A37-8, A42)
• Só podemos conhecer objetos que podemos, em princípio, intuir. Consequentemente, só podemos conhecer objetos no espaço e no tempo, nas aparências. Não podemos conhecer as coisas em si mesmos. (A239)
• No entanto, podemos pensar nas coisas por si mesmas usando as categorias (A254).
• As coisas em si nos afetam, ativando nossa faculdade sensível (A190, A387). [1]

Raymond Tallis on maths’ unreasonable effectiveness (excerpts)

Raymond Tallis sobre a eficácia irracional da matemática (trechos)

Talvez a maior evidência que a realidade não é matemática seja que o mundo das leis físicas – que permite previsões de quantidades – é um mundo de quantidades sem qualidades.

Isso não é um descuido acidental. Galileu, que deu início à revolução científica, argumentou que cores, gostos, sons, odores, não tinham lugar no mundo material, cujo livro foi escrito em matemática. As qualidades que experimentamos foram introduzidas por seres sencientes. Por outro lado, a própria realidade física era composta de ‘qualidades primárias’, como tamanho, forma, localização e movimento, que podem ser expressas em termos matemáticos sem deixar resíduos. Assim, o universo matemático da física carece do que é agora (depois de John Locke) geralmente chamado pelo termo levemente depreciativo “qualidades secundárias”. Estas resistem a serem matematizadas. A matemática da luz não chega nem perto da experiência do amarelo, nem a descrição matemática dos padrões dos impulsos nervosos atinge a própria dor. Às vezes, isso é visto como evidência de que nem a cor nem a dor são realmente reais – embora possa ser difícil vender essa afirmação para o homem que olha um narciso ou uma mulher com dor de dente.

E há algo mais que falta: atualidade. As leis físicas descrevem as relações mais gerais entre os padrões de mudança. Mas um padrão de relações entre eventos ou objetos não é um evento ou um objeto singular, realizado através das qualidades que os caracterizam. Wigner apontou algo nesse sentido quando observou que “as leis da natureza são todas declarações condicionais e se relacionam apenas a uma parte muito pequena de nosso conhecimento do mundo… [elas não dão] informações sobre a existência, as posições atuais, ou velocidades desses corpos ”- em resumo, sobre as realidades, todas fora das leis em si, sendo as condições nas quais as leis operam.

O mundo das leis físicas – que permite previsões de quantidades – é um mundo de quantidades sem qualidades.

Mesmo assim, ainda precisamos explicar a eficácia “irracional” da física e da matemática que está no seu coração. Claramente, a física deve estar acertando algo muito fundamental. Descartar os pitagóricos contemporâneos ao argumentar que a física matemática compreende apenas os aspectos quantitativos da realidade – daí suas previsões quantitativas extraordinariamente precisas – deixa inexplicável o fato de que essas previsões também permitem tecnologia que pode moldar o mundo – experiências reais, eventos, objetos e outras coisas – de acordo com nossos desejos. A matemática faz mais do que um contato passageiro com nossas vidas. Por outro lado, não podemos ignorar os outros tipos de verdades, enraizados na experiência real dos seres humanos que estão além da matemática: verdades situacionais saturadas de qualidades, sentimentos e preocupações, e diferenciações de espaço e tempo (‘aqui’, ‘agora ‘).

 

Géométrie et nombres

                            

Veja em Português        See it in English

Voir en premier: Réalité et chiffres

Ce  texte est librement soutenu par le texte de Painton Cowen et autres, mentionné.
J’ai déclaré dans l’entrée Design que:
«Peut-être que l’Univers est un nombre. Mais ce n’est pas pour moi.Je suis fermement convaincu que le nombre n’existe pas dans l’esprit du Créateur, il n’existe que dans la tête de l’homme. Les grands penseurs de l’antiquité croyaient que tout était un nombre parce qu’ils avaient l’illusion de pouvoir saisir complètement l’Univers dans une formule».

Les grands penseurs modernes ont intronisé cette idée comme étant la seule vraie.

Je, Roque, ne veux pas que mon affirmation de vider les nombres soit prise pour un mépris ignorant, ou pire encore, incompétent de ma part. Tout d’abord, je veux dire que j’aime le calcul, ou calculer, presque dans la même proportion que j’aime, par exemple, toute cette discussion sur Chartres. Aussi incroyable que cela puisse paraître à quiconque lit, la seule chose dans cette existence qui m’a vraiment donné suffisamment de gain pour dominer la vie et y mettre l’ordre significatif que je pensais (juste ou pas), était ma capacité à calculer, ce que je utilisé professionnellement chez IBM, où j’ai travaillé, en particulier lorsque je me suis impliqué dans la conception de diagnostics pour les ordinateurs.
On parle de géométrie et de nombres, mais en fait, la question est les mathématiques, dont la géométrie et les nombres sont un chapitre hautement expressif et qui dans notre cas sont fondamentales.
Une chose qui nous surprend lorsque nous abordons formellement le sujet est que dans les temples de la connaissance, c’est le cas de l’IMECC – Institut de mathématiques statistiques et d’informatique (UNICAMP), où j’ai étudié, cela existait et je crois toujours exister, deux courants par rapport aux mathématiques par rapport à la réalité:
Un groupe qui pense que les mathématiques ne devraient avoir aucun but pratique et devrait, en priorité ou peut-être exclusivement, être spéculatif et un autre groupe qui pense que les mathématiques devraient être utilisées pour explorer la réalité et aider d’autres sciences, que ce soit dans les mesures ou dans la conceptualisation.
C’est peut-être mon ignorance, mais je crois que la plupart des êtres humains partagent mon idée que les mathématiques doivent être pour conceptualiser le monde physique qui nous entoure et que, purement spéculatif, je ne dirai pas ce que je pense, car quand j’ai fait cela, lors d’une réunion départementale qui a jugé la faisabilité de créer un Master pour la Qualité parrainé par IBM, ce qui était un anathème pour ceux qui ne pensent qu’à la spéculation, j’ai créé des ennemis ou généré du mépris et des controverses dans une proportion que je considère absurde, pour ne pas dire quelque chose pire …
Un ordinateur est essentiellement une machine à calculer qui le fait inévitablement d’une manière mathématique. J’ai fait des incursions sur cet aspect et j’ai même importé 50 affiches «Men and Modern Mathematics», qui sont distribuées par divers instituts de mathématiques ou autres à travers le Brésil. Je développe séparément la soi-disant forme conventionnelle acceptée, en mathématiques, qui peut être examinée comme preuve que j’ai aussi une perception conventionnelle.
Comme il s’agit ici d’une façon de penser la réalité, je veux mettre en évidence les statistiques des mathématiques, car les statistiques ont une relation particulière avec les mathématiques. En général, les deux sont regroupés et nous avons presque toujours un «Département de mathématiques et de statistique». Parfois, les statistiques sont simplement considérées comme une branche des mathématiques appliquées. Les mathématiciens purs ont tendance à simplifier les statistiques comme une simple application de la théorie des probabilités, et la considèrent «pas suffisamment rigoureuse».
Je pense que la statistique est une branche des mathématiques qui utilise les mathématiques, mais qui diffère fondamentalement des autres branches des mathématiques, par exemple. analyse combinatoire, équations différentielles ou théorie des groupes. Les statistiques sont l’étude de l’incertitude, et cette incertitude est également l’objet des mathématiques, cependant, les mathématiques et les statistiques sont des modes de pensée fondamentalement différents.
L’axiome principal de la statistique est la fonction de Gauss ou la courbe Gaussienne et dans l’esprit d’accessibilité que j’essaie de donner au sujet, à quel point la distribution de l’intelligence est-elle normale?

Maintenant, pourquoi est-ce que je fais ce point? Premièrement, pour valider ma perception qu’il n’y a pas de nombre dans la nature (ou «dans l’esprit de Dieu…»), puisque probablement la seule justification qui me semble valable pour ce courant spéculatif est qu’en fin de compte, les mathématiques n’est efficace que spéculativement…
«Le livre sur la nature est écrit dans le langage des mathématiques», a déclaré Galileu. Son argument complet est en fait plus compliqué que cela, mais les philosophes, les scientifiques et les historiens ont profité de sa déclaration pour caractériser la «vraie» science: aucune branche de la philosophie naturelle ne pourrait être qualifiée de science avant qu’elle ne devienne mathématique. «Il n’y a que tellement de science authentique dans toute science, parce qu’elle contient des mathématiques», a déclaré Immanuel Kant au 18e siècle.
Cependant, Galilée a eu la chance de travailler dans la mécanique et l’astronomie: des problèmes qui peuvent être facilement mathématisés, mais ce sont peut-être ceux qui manquent le plus de modèles réels. En d’autres termes, ils sont toujours partiellement vrais. Même aujourd’hui, cette définition basée sur les mathématiques ne décrit toujours pas avec précision une grande partie de la chimie (c’est pourquoi Kant a nié que c’était de la science), de la biologie, des sciences de la terre ou de la médecine. Il n’existe pas d’équation unique dans «De l’origine des espèces» de Charles Darwin.

Et, surtout, l’explication «scientifique» qui compte est aujourd’hui, qui va évoluer en fonction du degré de précision que le progrès permet de rechercher et d’analyser.
Encadrant un peu mieux, les mathématiques surgissent effectivement avec les Grecs, en particulier Aristóteles et Pitágoras L’ensemble de ce qui était des mathématiques n’a plus de sens aujourd’hui, car après que Pierre de Fermat, René Descartes et Isaac Newton, sont entrés en scène, les problèmes et méthodes classiques ont été laissés pour compte. Bien sûr, Galileu, pratiquement inventant la science, était le grand organisateur. En fait, l’invention de la science avait pour première proposition, ou prototype de proposition, la mesure de l’Enfer, qui, bien sûr, n’existe pas en tant que telle, et est un exemple parfait de la façon dont vous pouvez génétiquement mathématiquement quelque chose qui pour tous des fins pratiques n’existent pas.
À propos de Pythagore et de Descartes, je veux observer:

Nombre métaphysique de Pythagore

cf. Encyclopédie Britannica

Le point de vue philosophique des premiers adeptes de Pythagore était que les caractéristiques plus profondes de la réalité sont déposées, d’une certaine manière, en nombres. La nature exacte de la relation de toutes choses avec les nombres, comme le voient les adeptes de Pythagore et lui-même, n’est pas claire. Pitagoras et ses premiers disciples évitèrent, en principe, de s’engager à écrire leurs conceptions, voire à enseigner. Les érudits ultérieurs, à ce jour, ont constaté qu’il est impossible de récupérer la pensée des philosophes présocratiques, car il n’y a pas de corpus de connaissances écrit ou décrit.
Comme le rapporte Aristote, les pythagoriciens ont été les premiers à faire des progrès dans l’étude des mathématiques et à y penser comme le principe de toutes choses. Selon Aristote, ils supposaient aussi que les éléments de nombres étaient aussi des éléments de choses et qu’ils existaient en imitant les nombres.
S’il est clair que d’une manière ou d’une autre les mathématiques ont gouverné sa position métaphysique, les chercheurs ultérieurs ont été incapables de reconstruire les théories de base qui étayaient ces affirmations.
Ils étaient presque spéculatifs.

La métaphysique comme science a priori – Descartes et l’utilisation des principes de la géométrie pour cela

cf. Encyclopédie Britannica

Spinoza a pris de Descartes la conception de la connaissance avec laquelle il affirme, selon la géométrie, qu’il y a des choses évidentes en éthique et de huit définitions et sept axiomes, il a créé 36 propositions qui forment la base de son texte sur l’éthique.
Descartes, qui a également traité de la possibilité de quelque chose dans ce sens, c’est-à-dire de présenter des arguments métaphysiques de manière géométrique, a souligné que, bien qu’il n’y ait pas de difficulté avec les premiers principes de la géométrie, «rien en métaphysique ne pose plus de problèmes que de faire vos bases clair et distinct. ”
Tout le problème de cette discipline réside dans le fait que les étudiants ne se rendent pas compte qu’ils doivent partir de ce qui sont en fait ses vérités fondamentales.
Descartes lui-même a parlé comme s’il ne s’agissait que d’un problème pédagogique, juste une question de faire voir comme une évidence ce qui en soi va de soi.
Descartes était optimiste, car la difficulté de son système et ce que Spinoza a créé avec ses idées est qu’il y a des gens qui ne peuvent pas voir, même s’ils essaient ou leur enseignent, que les propositions de base du système sont évidentes.
Cela suggère que dans tout système de ce type, il est nécessaire d’avoir un élément arbitraire ou du moins non obligatoire.
Je fais partie de ces personnes, je le vois comme ça et comme le disent les avocats américains «je repose ma cause», c’est-à-dire que je considère mon argument défendu, c’est-à-dire qu’il n’y a aucune preuve fiable que la réalité est inextricablement basée sur des chiffres.
J’ajouterais également que les similitudes et la ressemblance sont beaucoup plus un cas du point de vue et de l’exactitude de l’observateur, c’est-à-dire une limite de perception de ce qui est constitué ou comment il est apparu ou comment il existe sur ce qui est considéré. .
Qu’est-ce que je pense de toute façon?
Cette réalité est un mystère qui s’est lentement dévoilé dans plusieurs de ses dimensions pour l’homme, mais qu’elle ne sera jamais complètement dévoilée, restant… un mystère!

Géométrie et gothique

La géométrie considère les formes, à la fois dans le plan et dans les solides. En effet, les grandes rosaces de Paris et de Chartres qui nous coupent le souffle, avec leurs spectaculaires toiles de verre, d’étain et de pierre, sont une perfection qui prend forme à travers la géométrie.
Le transept nord de Chartres est construit selon trois ensembles géométriques superposés, dont l’un est basé sur le suite de Fibonacci.
Une deuxième caractéristique géométrique incorpore toutes les caractéristiques principales du vitrail dans un système de triangles équilatéraux.
Une troisième caractéristique est obtenue par la composition des deux premiers.
De plus, le numéro 12 se trouve sous tout le vitrail, en tant que produit 3 par 4, symbolisant respectivement la Trinité et les quatre éléments – terre, air, eau et feu. Les combinaisons de 4 et 3 sont 12, symbolisant l’infusion totale de la matière avec l’esprit à travers le cosmos – dont la rosace est un modèle!
De cette manière, l’action de l’Esprit peut être vue dans le vitrail comme une colombe à quatre dimensions, puisque l’Esprit agit continuellement à travers le temps et l’espace – «aimant et guidant toute matière créée», comme le dit l’auteur Thierry. Le symbole de cet amour est Eros, ou l’Amour Divin du pouvoir créateur du Cosmos. C’est une fusion parfaite et une manifestation de la philosophie de l’École de Chartres: quelque chose d’une grande beauté créé par l’homme pour Dieu.

Dans le vitrail oriental de Laon, le sens implicite est caché d’une autre manière. Géométriquement, le vitrail est construit avec douze pentagones qui entourent le centre – un écho du cinquième solide platonique, le dodécaèdre, dont les douze faces sont des pentagones. Dans son œuvre Timée, les quatre éléments, feu de terre, air et eau sont représentés par le cube, la pyramide, l’octaèdre et l’icosaèdre (20 côtés). Ces solides corroborent la matière, qui se manifeste avec le temps. Le cinquième élément, l’éther, peut être considéré comme représenté par le dodécaèdre «le ciel spirituel complet», qui est éternellement manifesté. En cela, l’iconographie en rosace de Laon confirme la représentation de l’imagerie chrétienne comme le mystère éternel, la jonction de l’ancien et du nouveau.
Il y a une implication tridimensionnelle dans toutes les rosaces du vitrail gothique et le dodécaèdre de Laon en est un bon exemple.
D’autre part, l’aspect mandala de toute rosette nous aide à comprendre cette tridimensionnalité et suggère une signification «cosmique»: la forme bidimensionnelle représentant la forme tridimensionnelle qui par notre imagination (sens du mandala) nous élève à une réalité à quatre dimensions

Nombres et géométrie en création

Chaque rosace gothique est une expression directe du nombre et de la géométrie – d’une lumière parfaitement formée. A Chartres toutes les rosaces sont divisées en 12 segments, nombre de perfection, de l’univers et du Logos. Les chercheurs de Chartres étaient clairement fascinés par les nombres et la géométrie qui en dérivait, non pas comme un moyen en soi mais comme une clé pour comprendre la nature. Ils ont étudié Pitagoras pour qui, selon la tradition, la géométrie était divine et les nombres éternels – puisque tout périt (selon ma théorie, les nombres n’existent pas….).
La fascination pour les nombres était déjà entrée à Chartres lorsque Sto. Augustin, qui a également vu la sagesse divine se refléter dans les nombres imprimés sur toutes choses.
Les nombres et la géométrie représentent l’ordre, le mot grec kosmos signifie l’ordre, donc l’étude du cosmos impliquait une étude des nombres.
Je fais une parenthèse pour défendre mon point.

Le chaos, la destruction, le désordre, la folie ne font partie d’aucune idée largement acceptée comme bonne pour comprendre le monde. Il y a des tentatives qui, sans succès, ont laissé une impression sur l’imagination humaine. Par exemple, les conceptions de Marques de Sade, le nazisme. Cependant, ils ne l’ont pas eux-mêmes identifié comme chaotique ou mauvais. Ils ont défendu quelque chose qui leur a semblé un «bien». L’ordre qui existe dans le monde est indéniable, tout comme le désordre est indéniable. Le mal lui-même, qui serait la synthèse du contraire de l’ordre, est défini comme l’absence de bien et n’a pas de définition propre. J’insiste sur le fait que notre équipement mental est à un niveau conscient construit pour reconnaître l’ordre et référencer tout ce qui s’y rapporte. Bien sûr, le nombre reflète l’ordre, cependant, car je pense que la réalité est 49,9999999999999999999% mauvais et 50,000000000000000000000001 bien, environ la moitié de tout ce qui existe, qui à mon avis existe toujours coincé dans un processus paradoxal avec son contraire, est sans ordre «numérique ”Critère de compréhension. Je ne suis pas familier avec des concepts similaires à ce dont nous discutons, mais inversement, c’est-à-dire faire face à l’opposé du bien et du beau. Je crois qu’il serait possible, par exemple, de construire une cathédrale «noire», avec tout ce qui est noir, y compris des vitraux et des mandalas «du mal». Ce serait une chose folle. Mais pour comprendre la réalité, et pour justifier ce point des chiffres, il faudrait qu’elle existe.
Comme notre équipement mental ne changera pas en substance, il est interdit à l’homme de tout comprendre. Et ce qu’il comprend est lié à un processus que les Grecs reflètent à merveille dans plusieurs mythes, dont je souligne  Cupido e Psiche,  Sisifo et  Danaan.

C’est pourquoi je choisis le mystère.

Je veux rappeler à ceux qui ont des illusions scientifiques que, jusqu’à il y a quelque temps, la terre était le centre de l’univers et très récemment, sur la base des idées d’Einstein, il est strictement scientifique de voyager dans l’espace et de revenir avant la naissance des parents et les grands-parents… ce qui est clairement pour moi le reflet d’une idée «ordonnée» qu’avait Einstein et qui reflète certes beaucoup de réalité. Mais la réalité doit être autre chose aussi, pour ajuster ce paradoxe, qui est le nom scientifique de ce phénomène.
De retour à Chartres, on se souvient que les églises chrétiennes ont été construites selon des principes géométriques depuis les débuts de la chrétienté. Les chiffres étaient une préoccupation médiévale. La géométrie et l’arithmétique étaient des études traditionnelles, mais avec la découverte des «Eléments d’Euclide» par Abelardo de Bath, qui était lié à l’école de Chartres, un nouvel enthousiasme s’est manifesté.

Les éléments d’Euclide sont un recueil des concepts mathématiques de Pythagore organisés de manière systématique. C’était la première édition en anglais et l’un des textes les plus importants et les plus influents de l’ère élisabéthaine, qui a révolutionné la science et la technologie, en particulier le domaine de la navigation. Je n’ai pas évoqué le «fonctionnement du sextant», mais la base scientifique est pythagoricienne. Cette édition comprend la première apparition de la préface influente de l’alchimiste John Dee, qui défend l’utilisation pratique du texte d’Euclide. Ce livre est une étape importante dans la tâche d’impression (ou art) du 16ème siècle, non seulement pour la splendide allégorie de la page de titre, la xylographie du portrait de John Day, qui était l’imprimeur, et surtout, pour l’impression de 59 trois géométriques -des découpes dimensionnelles, qui sont parmi les premières tentatives jamais faites pour reproduire des solides géométriques dans un livre imprimé.
De plus, les nombres au Moyen Âge avaient acquis une signification métaphysique en eux-mêmes et, selon l’auteur Mäle, ils étaient imaginés comme s’ils avaient des pouvoirs cachés. Pour cette raison, ils ont été intégrés dans tous les aspects de la construction de la cathédrale, du nombre de piliers dans le chœur à la relation entre les niveaux du triforium et la disposition de la façade. Inévitablement, cela se reflétait plus que tout dans les rosaces.
Quant à la relation numérique, pour le vitrail, le chiffre 8 était le plus important, avec le plus important, le 12, et chacun avait un équivalent géométrique. Regardons chacun d’eux:

• le nombre 1 représentait l’unité de toutes choses, et son équivalent était le cercle et son centre,
• le nombre 2 représentait la dualité du paradoxe des contraires, exprimé par paires le long du centre,
• le nombre 3, le triangle, la stabilité de la transcendance de la dualité,
• le chiffre 4, le carré, la matière, les éléments, les vents, les saisons, les directions,
• le chiffre 5, le pentacle, l’homme, la magie et le Christ crucifié avec cinq blessures,
• le nombre 6, le nombre d’équilibre et d’harmonie dans l’âme, symbolisé par l’étoile de David ou le sceau de Salomon,
• le chiffre 7, est un nombre mystique, représentant les sept âges, les planètes, les vertus, les dons de l’esprit et les arts libéraux,
• le nombre 8, est le nombre de baptême et de renaissance, implicite dans l’octogone
  le nombre 9, est le numéro de l’achèvement du travail humain
• numéro 10, est le numéro d’achèvement de la commande, il ne manque plus rien
• le nombre 11, est le numéro de quelque chose d’autre lorsque tout est terminé, par exemple. le livre de l’Apocalypse
• le nombre 12 (et le 24), sont les nombres les plus courants dans les rosettes, en particulier dans les transepts orientés au sud
• Les rosaces en 5 ou 8 parties et leurs multiples sont généralement orientées au nord, mais il n’y a pas de règles strictes.

A Notre Dame, Paris, il y a 16 énormes pétales sur la face nord, 12 au sud, la même chose se passe à Rouen. A Clermont-Ferrand et à Tours, il y a des rosaces à 16 pétales dans les deux transepts. Beaucoup plus convaincant quant au placement de rosaces à 5 pétales au nord face à 6 ou 12 au sud est le cas d’Amiens, Sens et Saint-Oen à Rouen.
Avant d’aller plus loin dans la géométrie, pour ceux qui sont curieux de savoir, visitez les sites de signification des nombres:

• signification des nombres à l’époque pythagoricienne 
• signification des nombres comme symboles
• signification des nombres dans la Bible
• un autre de la Bible 
• quelle est la signification des nombres en numérologie
• nombres dans les rêves
• quelle est la signification des nombres dans les symboles au Moyen Âge
• chiffres du point de vue de la science moderne 
• quelle est la signification des nombres dans le livre de l’Apocalypse
• nombres et religion
• les maths que tout le monde devrait savoir 

En conclusion:

Les vitraux gothiques utilisent la géométrie (et les nombres) de trois manières différentes:

• Une manière manifeste ou explicite
• Un caché et
• Une manière symbolique

Ces trois voies peuvent être reliées, reliant le symbolique au réel dans la description de l’ordre créateur du Logos. Otto von Simson souligne que c’est plus dans la conception que dans les mesures que la géométrie est généralement utilisée dans les cathédrales gothiques. Cela reflète la vérité que dans le monde des nombres, tout est relatif. En ce sens, la rosace devient symbolique de l’infini, sans dimension et englobe tout, comme l’Amour du Créateur, qui agit de l’atome à la galaxie. Cela a quelque chose de la représentation islamique de Dieu sous forme géométrique, qui à travers un tissu infini exprime sa créativité infinie.
Dans le vitrail gothique, c’est principalement le sens manifeste et explicite qui donne le premier impact visuel et le réseau de complexité et de précision qui se trouve dans chaque espace, défini par des figures géométriques encore plus petites, telles que trois pétales, quatre pétales, des rosaces ou sphériques. Triangles. Ajouté à cela, un métier à tisser tisse un tissu lacé de mosaïques rouges et bleues, symbolisant l’activité du Mot Créateur jusqu’à la plus petite fibre dans le plus petit coin de la rose cosmique.
Sous cette structure visible, c’est la géométrie tout aussi précise. Dans la plus grande rosace, la géométrie définit la position exacte de presque toutes les caractéristiques principales, reliant les éléments radiaux aux divisions concentriques et tout avec le centre. Notre œil, peut-être inconsciemment, perçoit ces relations de la même manière que la géométrie est cachée tout au long de la construction du bâtiment. L’auteur Villard de Honnencourt montre de nombreuses formes géométriques dans ses dessins schématiques sous forme d’animaux, d’humains et du bâtiment lui-même. L’intérêt de cet auteur pour les triangles en tant qu’élément de base est le reflet des idées de Platon exprimées à Timée. Il est intéressant de noter que ce livre de Villar de Honnencourt contient deux schémas de rosaces, l’un de Chartres et l’autre de Lausanne, les deux, pour des raisons inconnues, sont inexacts.
Dans de nombreuses religions du monde, la géométrie symbolique est utilisée comme un type de raccourci: les cercles, les carrés, les triangles et les étoiles sont utilisés pour incorporer un sens beaucoup plus profond que ce qu’ils ont, par exemple, dans les panneaux de signalisation.
Les carrés et les cercles ont une signification universelle symbolisant le fini et l’infini, le ciel et la terre ou la matière et l’esprit.
Dans la pensée soufie, le centre, le rayon et la circonférence d’un cercle symbolisent respectivement la Vérité, la Voie et la Loi. Le centre est alors le point d’équilibre, le point au repos, ou le point d’intersection intemporelle avec le temps, selon les mots de TSElliot.
Dans de nombreuses rosaces, le fini et l’infini s’unissent en plaçant la partie circulaire avec le carré, comme à Paris, Clermont-Ferrand, Séese et bien d’autres. Lorsque le nombre 12 est défini dans la rosace, l’union du ciel et de la terre est symboliquement complète. A Lausanne, la combinaison est explorée à fond.
Ces rosaces et vitraux mentionnés ci-dessus peuvent être vus dans Exemples de vitraux gothiques

Depuis les temps les plus reculés, le cercle signifie l’éternité, Dieu, la vénération, la perfection, l’année et le ciel. Et pour cette raison, selon Vitruve, que certains temples antiques sont ronds, représentant la forme ou la figure du ciel. En tant que symbole du Temple de l’Esprit dans l’homme, la rose symbolise l’unité de toutes choses, lorsque les contraires sont réconciliés par le centre. Une petite rosace dans le choeur de la cathédrale d’Auxerre a les vices et les vertus disposés en opposition les uns aux autres, comme les signes antithétiques, c’est-à-dire en antithèse, du zodiaque. Sto Tomas de Aquino à la fin du 13ème siècle a suivi Aristoteles dans le concept de Le nombre d’or , également connu sous le nom de Segment d’or ou extrême raison, comme par exemple le nombre Phi. Cela a été présenté comme une troisième possibilité, authentique, sans dualité.
La foi et la raison étaient des opposés irréconciliables de l’époque médiévale, du moins jusqu’à saint Thomas d’Aquin. Cependant, avant de trouver une solution, les constructeurs de cathédrales mettaient en œuvre leurs versions pour résoudre le même problème d’intégration des contraires, la rosacée, qui était un type de bouée de sauvetage dans les eaux confuses de l’époque.
Plusieurs années plus tard, S.João da Cruz a verbalisé le problème des contraires et du paradoxe, ce qui a fait écho à TS Elliot dans les mots de ses vers:

“Pour arriver où tu es, pour arriver d’où tu n’es pas,
Vous devez suivre une voie où il n’y a pas d’extase.
Pour arriver à ce que vous ne savez pas
Vous devez suivre une voie qui est la voie de l’ignorance.
Afin de posséder ce que vous ne possédez pas
Vous devez suivre la voie de la dépossession.
Afin d’arriver à ce que vous n’êtes pas
Vous devez passer par la manière dont vous n’êtes pas.
Et ce que tu ne sais pas est la seule chose que tu sais
Et ce que vous possédez est ce que vous ne possédez pas
Et là où vous êtes, c’est là où vous n’êtes pas.”

Enfin, si TS Eliot n’avait rien écrit, le verset ci-dessus suffirait à le mettre là où il est.
Voir les schémas géométriques

Voir les schémas géométriques

Retour au design

Geometry and Numbers

                               

Veja em Português        Voir en français

See first: Reality and Numbers

This text is freely supported by the  text of Painton Cowen and others, mentioned.
I stated in the Design entry that:
“Perhaps the Universe is number. But it is not for me. I am strongly convinced that number does not exist in the Creator’s Mind, it does only in the head of man. The great thinkers of antiquity believed that everything was a number because they had the illusion of being able to completely grasp the Universe in some formula ”.

The great modern thinkers have enthroned this idea as the only true one.

I, Roque, do not want my assertion in some way emptying numbers to be mistaken for some ignorant contempt, or even worse, incompetent, on my part. First of all, I want to state that I like calculus, or calculate, almost in the same proportion that I like, for example, this whole discussion about Chartres. Incredible as it may seem to anyone who reads, the only thing in this existence that really gave me ample gain enough to dominate life and put in it the meaningful order that I thought (right or not), was my ability to calculate, which I used professionally at IBM, where I worked, especially when I got involved in   diagnostics design for computers.
We are talking about geometry and numbers, but in fact, the question is mathematics, of which geometry and numbers are a highly expressive chapter and which in our case are fundamental.
One thing that surprises us when we formally approach the subject is that within temples of knowledge, such is the case with IMECC – Institute of Statistical Mathematics and Computer Science (UNICAMP), where I studied, it existed and I believe still exist, two currents with respect to mathematics in relation to reality:
A group that thinks that mathematics should have no practical purpose and should, as a priority or perhaps exclusively, be speculative and another group that thinks that mathematics should be used to explore reality and help other sciences, whether in measurements or in conceptualization.
It may be my ignorance, but I believe that most human beings share with my idea that mathematics has to be to conceptualize the physical world that surrounds us and that, purely speculative, I will not say what I think, because when I did this , in a departmental meeting that judged the feasibility of creating a Master’s Degree for Quality that IBM sponsored, which was anathema to those who think only of speculation, I created enemies or generated contempt and controversies in a proportion that I consider absurd, not to say something worse …
A computer is essentially a calculating machine that inevitably does so in some mathematical way. I made inroads on this aspect and even imported 50 posters “Men and Modern Mathematics, which is distributed by various Institutes of Mathematics or the like throughout Brazil. I elaborate separately on the so-called conventional accepted form, in mathematics, which can be examined as evidence that I have conventional perception as well.
As the point here is a way of thinking about reality, I want to highlight the statistics of mathematics, because statistics have a peculiar relationship with mathematics. In general, the two are grouped and we almost always have a “Department of Mathematics and Statistics”. Sometimes, statistics are just regarded as a branch in applied mathematics. Pure mathematicians tend to simplify statistics as just an application of the theory of probability, and consider it “not sufficiently rigorous”.
I believe that statistics is a branch of mathematics that uses mathematics, but differs fundamentally from other branches of mathematics, for example, combinatorial analysis, differential equations or group theory. Statistics is the study of uncertainty, and that uncertainty is also the object of mathematics, however, mathematics and statistics are fundamentally different ways of thinking.
The main axiom of Statistics is the Gauss Function or the Gauss Curve and within the spirit of accessibility that I am trying to give to the subject, How normal is the distribution of intelligence
Now, why do I make this point? First, to validate my perception that there is no number in nature (or “in the Mind of God…”), since probably the only justification that seems to me valid for this speculative current is that,  at the end of the day, mathematics is effective only speculatively…
The nature book is written in the language of mathematics, ”said Galileo. His full argument is actually more complicated than that, but philosophers, scientists and historians took advantage of his statement to characterize “true” science: no branch of natural philosophy could be called a science until it became mathematics. “There is only so much genuine science in any science, because it contains mathematics,” said Immanuel Kant in the 18th century.
However, Galileo was fortunate to work in mechanics and astronomy: problems that can be conveniently mathematized, but perhaps are the ones that most fail to have real models. In other words, they are always partially true. Even today, that definition based on mathematics still does not accurately describe much of chemistry (which is why Kant denied it was science), biology, earth science or medicine. There is no single equation in Charles Darwin’s “On the Origin of Species”.

And, above all, the “scientific” explanation that counts is today, which will change depending on the degree of accuracy that progress allows for research and analysis.
Framing it a little better, mathematics actually arises with the Greeks, especially  Aristóteles and Pitágoras The whole of what was mathematics makes no more sense today, because after  Pierre de Fermat, René Descartes and Isaac Newton, entered the scene, the classic problems and methods were left behind. Of course, Galileo, practically inventing science, was the great organizer. In fact, the invention of science, had as its first proposal, or prototype of a proposal, as the measurement of Hell, which of course does not exist as such, and is a perfect example of how you can genetically mathematicize something that for all practical purposes does not exist .
About Pythagoras and Descartes I want to observe:

Metaphysical Number of Pythagoras

cf. Encyclopaedia Britannica

The philosophical view of the early followers of Pythagoras was that the deeper characteristics of reality are deposited, in some way, in numbers. The exact nature of the relationship of all things to numbers, as seen by the followers of Pythagoras and himself, is unclear. Both Pitagoras and his early followers, avoided, in principle, the commitment to write their conceptions, even to teach. Later scholars, to date, have found that it is impossible to recover the thought of pre-Socratic philosophers, since there is no body of knowledge written or described.
As reported by Aristotle, the Pythagoreans were the first to make progress in the study of mathematics and to think about it as the principle of all things. According to Aristotle, they also assumed that the elements of numbers were also elements of things and that they existed by imitating numbers.
While it is clear that somehow mathematics governed its metaphysical position, later scholars were unable to reconstruct the basic theories that were substantiating these claims.
They were almost speculative.

Metaphysics as an a priori science – Descartes and the use of the principles of geometry for this

cf. Encyclopaedia Britannica

Spinoza took from Descartes the conception of knowledge with which he, according to geometry, affirms that in Ethics there are self-evident things and from eight definitions and seven axioms, he created 36 propositions that form the basis of his text on Ethics.
Descartes, who also dealt with the possibility of something along these lines, i.e., presenting metaphysical arguments in a geometric way, stressed that, although there was no difficulty with the first principles of geometry, “nothing in metaphysics causes more problems than making your basics clear and distinct. ”
The whole problem with this discipline lies in the fact that students do not realize that they need to start from what are in fact its basic truths.
Descartes himself spoke as if it was just a pedagogical problem, just a matter of making people see as self-evident what in itself is self-evident.
Descartes was optimistic, because the difficulty of his system and what Spinoza created with his ideas is that there are people who cannot see, however much they try or teach them, that the basic propositions of the system are self-evident.
This suggests that in any system of this type there is a need for an arbitrary or at least non-compulsory element.
I am among these people, I see it like this and as American lawyers say “I rest my case”, that is, I consider my point defended, that is, there is no reliable evidence that reality is based on numbers inextricably.
I would also add that similarities and likeness are much more a case of the observer’s point of view and accuracy, that is, a limit of perception of what is constituted or how it came up to exist or how it does exist on what is being considered.
What do I think anyway?
That reality is a mystery that has slowly unveiled itself in many of its dimensions for man, but that it will never be fully unveiled, remaining … a mystery!

Geometry and Gothic

Geometry considers forms, both in the plane and in solids. In fact, the great rosettes of Paris and Chartres that take our breath away, with their spectacular webs of glass, tin and stone, are a perfection that takes shape through geometry.
The northern transept of Chartres is built according to three overlapping geometric sets, one of which is based on the  Fibonacci series.
A second geometric feature incorporates all the main characteristics of the stained glass window within a system of equilateral triangles.
A third characteristic is obtained by the composition of the first two.
In addition, the number 12 is under the entire stained glass window, as a 3 by 4 product, respectively symbolizing the Trinity and the four elements – earth, air, water and fire. The combinations of 4 and 3 are 12, symbolizing the total infusion of matter with the spirit through the cosmos – of which rosette is a model!
In this way, the Spirit’s action can be seen in the stained glass window as a four-dimensional dove, since the Spirit acts continuously through time and space – “loving and guiding all created matter”, as the author Thierry puts it. The symbol of this love is Eros, or the Divine Love of the creative power of the Cosmos. It is a perfect fusion and manifestation of the philosophy of the School of Chartres: something of great beauty created by man for God.

In Laon’s east stained glass the implicit meaning is hidden in another way. Geometrically, the stained glass is constructed with twelve pentagons that surround the center – an echo of the fifth platonic solid, the dodecahedron, whose twelve faces are pentagons. In his work Timaeus, the four elements, earth fire, air and water are represented by the cube, the pyramid, the octahedron and the icosahedron (20 sides). These solids substantiate matter, which manifests itself over time. The fifth element, ether, can be said to be represented by the dodecahedron “the complete spiritual sky”, which is eternally manifest. In this, Laon’s rosette iconography confirms the representation of the Christian imagery as the Eternal Mystery, the junction of the Old and the New.
There is a three-dimensional implication in all the rosettes of the Gothic stained glass and the Laon dodecahedron is a good example.
On the other hand, the mandala aspect of any rosette helps us to understand this three-dimensionality and suggests a “cosmic” meaning: the two-dimensional shape representing the three-dimensional shape that by our imagination (meaning of the mandala) elevates us to a four-dimensional reality

Numbers and Geometry in creation

Every Gothic rose window is a direct expression of number and geometry – of perfectly shaped light. In Chartres all rosettes are divided into 12 segments, number of perfection, of the universe and of the Logos. Chartres scholars were clearly fascinated by numbers and the geometry derived from them, not as a means in itself but as a key to understanding nature. They studied Pitagoras for whom, according to tradition, geometry was divine and numbers eternal – since everything perishes (according to my theory, numbers do not exist….).
Fascination with numbers had already entered Chartres when Sto. Augustine, who also saw divine wisdom reflected in numbers printed on all things.
Numbers and Geometry represent order, the Greek word kosmos means order, so the study of the cosmos involved a study of numbers.
I make a parenthesis to defend my point.

Chaos, destruction, disorder, madness are not part of any idea that is widely accepted as good for understanding the world. There are some attempts that, without being successful, left an impression on the human imagination. For example, Marques de Sade’s conceptions, Nazism, etc. However, they themselves did not identify it as chaotic or evil. They defended something that seemed to them as “good”. The order that exists in the world is undeniable, as disorder is undeniable. Evil itself, which would be the synthesis of the opposite of order, is defined as the absence of good and has no definition of its own. I insist that our mental equipment is at a conscious level built to recognize order and reference everything in relation to this. Of course, the number reflects order, however, as I believe the reality is 49.9999999999999999999% bad and 50.000000000000000000000001 well, approximately half of everything that exists, which in my view always exists stuck in a paradoxical process with its opposite, is without ordered “numeric” criterion of understanding. I am not familiar with concepts similar to what we are discussing, but conversely, that is, facing the opposite of good and beautiful. I believe it would be possible, for example, to build a “black” cathedral, with everything black, including stained glass and mandalas “from evil”. It would be a crazy thing. But to understand reality, and to justify this point of the numbers, it would have to exist.
As our mental equipment will not change in essence, it is forbidden for man to understand everything. And what he understands is tied to a process that the Greeks reflected wonderfully in several myths, of which I highlight Cupido e Psiche,  Sisifo and  Danaan.

That is why I choose mystery.

I want to remind those who have scientific illusions that, until some time ago, the earth was the center of the universe and very recently, based on Einstein‘s ideas, it is strictly scientific to travel into space and return before the birth of parents and grandparents … which clearly for me is a reflection of an “orderly” idea that Einstein had and that admittedly reflects a lot of reality. However, reality has to be something else too, to adjust this paradox, which is the scientific name of this phenomenon.
Returning to Chartres, we remember that the Christian churches were built according to geometric principles since the early days of Christendom. Numbers were a medieval concern. Geometry and Arithmetic were traditional studies, but with the discovery of the “Elements of Euclid” by Abelardo de Bath, who was linked to the Chartres School, a new enthusiasm arose.

The Euclid’s Elements is a compendium of the mathematical concepts of Pythagoras organized in a systematic way. It was the first edition in English and one of the most important and influential texts of the Elizabethan era, which revolutionized science and technology, particularly the field of navigation. I did not mention the “sextant functioning”, but the scientific basis is Pythagorean. This edition includes the first appearance of the influential preface by the alchemist John Dee, who defends the practical use of Euclid’s text. This book is a milestone in the 16th century printing task (or art), not only for the splendid allegory of the title page, the xylography of John Day’s portrait, who was the printer, and above all, for the printing of 59 geometric three-dimensional cutouts, which they are among the first attempts ever made to reproduce geometric solids in a printed book.
Furthermore, numbers in the Middle Ages had acquired a metaphysical significance in themselves and, according to the author Mäle, they were imagined as if they had hidden powers. For this reason, they were embedded in all aspects of the construction of the Cathedral, from the number of pillars in the choir to the relationship between the levels of triforium and the layout of the facade. Inevitably, this was reflected more than anything in the rose windows.
As for the numerical relationship, for stained glass, the number 8 was the most important, along with the most important, the 12, and each had a geometric equivalent. Let’s look at each one:

• the number 1 represented the unity of all things, and its equivalent was the circle and its center,
• the number 2 represented the duality of the paradox of opposites, expressed in pairs along the center,
• the number 3, the triangle, the stability of the transcendence of duality,
• the number 4, the square, the matter, the elements, the winds, the seasons, the directions,
• the number 5, the pentacle, the man, the magic and Christ crucified with five wounds,
• the number 6, the number of balance and harmony within the soul, symbolized by the star of David or the seal of Solomon,
• the number 7, is a mystical number, representing the seven ages, the planets, the virtues, the gifts of the spirit and the liberal arts,
• the number 8, is the number of baptism and rebirth, implicit in the octagon
• the number 9, is the number of the completion of human work
• number 10, is the order completion number, nothing more is missing
  the number 11, is the number of the something else when everything is complete, for example. the book of Revelation
• the number 12 (and the 24), are the most common numbers in rosetes, especially in south-facing transepts
• 5-part or 8-part rosettes and their multiples are generally facing north, but there are no strict rules.

At Notre Dame, Paris, there are 16 huge petals on the north face, 12 in the south, the same happening in Rouen. At Clermont-Ferrand and Tours, there are rosettes with 16 petals in the two transepts. Much more convincing regarding the placement of 5-petalled rosettes in the north facing 6 or 12 in the south is the case of Amiens, Sens and St.Oen in Rouen.
Before we go any further in geometry, for those who are curious to know, visit the number meaning sites:

• meaning of numbers in the Pythagorean era
• meanings of numbers as symbols
• meaning of numbers in the Bible
• another from the Bible
•  what is the meaning of numbers in numerology
• numbers in dreams
• what is the meaning of numbers in symbols in the Middle Ages
• na visão da ciência moderna
• what is the meaning of numbers in the book of Revelation
• numbers and religion
• Math everyone should know 

Concluding:

The gothic stained glass windows use geometry (and numbers) in three different ways:

• A manifest or explicit way
• A hidden and
• A symbolic way

These three ways can be connected, linking the symbolic with the real in the description of the creative order of the Logos. Otto von Simson points out that it is more in design than in measurements that geometry is generally used in Gothic cathedrals. This reflects the truth that in the world of numbers everything is relative. In this sense, the rosette becomes symbolic of the infinite, dimensionless and encompasses everything, like the Love of the Creator, which acts from the atom to the galaxy. This has something of the Islamic representation of God in geometric form, which through infinite fabric expresses His infinite creativity.

In the Gothic stained glass it is primarily the manifest and explicit sense that gives the first visual impact and the web of complexity and precision which is within each space, defined by even smaller geometric figures, such as three petals, four petals, rosettes or spherical triangles. Added to this, a loom weaves a fabric laced with red and blue mosaics, symbolizing the activity of the Creative Word down to the smallest fiber in the smallest corner of the cosmic rose.
Under this visible structure it is the equally precise geometry. In the largest rose window, the geometry defines the exact position of almost all the main characteristics, relating the radial elements to the concentric divisions and everything with the center. Our eye, perhaps unconsciously, perceives these relationships in the same way that geometry is hidden throughout the construction of the building. The author Villard de Honnencourt shows many geometric shapes in his schematic drawings in the shapes of animals, humans and the building itself. This author’s interest in triangles as a basic component is a reflection of Plato’s ideas expressed in Timaeus. It is interesting to note that this book by Villar de Honnencourt contains two rosette schemes, one by Chartres and the other by Lausanne, both, for unknown reasons are inaccurate.
In many religions of the world symbolic geometry is used as a type of shorthand: circles, squares, triangles and stars, are used to incorporate a much deeper sense than they have, for example, in traffic signs.

Squares and circles have universal significance symbolizing finite and infinite, heaven and earth or matter and spirit.
In Sufi thought, the center, radius and the circumference of a circle respectively symbolize Truth, the Way and the Law. The center is then the point of equilibrium, the point at rest, or the point of the timeless intersection with time, in the words of TSElliot.
In many rosettes, the finite and the infinite are united by placing the circular part with the square, as in Paris, Clermont-Ferrand, Séese and many others. When the number 12 is defined within the rosette, the union of heaven and earth is symbolically complete. In Lausanne the combination is explored to the full.
These rosaceas and stained glass mentioned above can be seen in Examples of Gothic Stained Glasses

Since the earliest times the circle has meant eternity, God, veneration, perfection, the year and the sky. And for this reason, according to Vitruvius, that some ancient temples are round, representing the shape or figure of the sky. As a symbol of the Temple of the Spirit within man, the rose symbolizes the unity of all things, when opposites are reconciled through the center. A small rose window in the choir of the Cathedral of Auxerre has the vices and virtues arranged in opposition to each other, like the antithetical signs, that is, in antithesis, of the Zodiac. Sto Tomas de Aquino at the end of the 13th century followed Aristoteles in the concept of Golden Division, also known as Golden Segment or Extreme Reason, as for example the number Phi. This was presented as a third possibility, genuine, free from duality.
Faith and reason were irreconcilable opposites of the Medieval Age, at least until St. Thomas of Aquino. However, before he came up with a solution, cathedral builders were implementing their versions to solve the same problem of integrating opposites, rosacea, which was a type of lifesaver in the confused waters of the time.
Many years later, S. João da Cruz verbalized the problem of opposites and paradox, which made TS Elliot echo in the words of his verses:

“To arrive where you are, to get from where you are not,
You must go by a way wherein there is no ecstasy.
In order to arrive at what you do not know
You must go by a way which is the way of ignorance.
In order to possess what you do not possess
You must go by the way of dispossession.
In order to arrive at what you are not
You must go through the way in which you are not.
And what you do not know is the only thing you know
And what you own is what you do not own
And where you are is where you are not.”

Finally, if TS Elliot had not written anything, the verse above would be enough to put it where he is.

See Geometric Schemes

Back to Design

Le désign

                            

Veja em Português        See it in English

Le désign ou la conception du vitrail gothique

Pourquoi «Design»? Parce que ce n’est ni dessin ni projet. C’est la conception totale sous tous ses angles et dimensions. Un aspect très commenté des vitraux et de la conception en gothique et cathédrale gothique par extension, est la question de la Géométrie, des nombres, des mesures, du plan, des mathématiques en somme.

Je ne veux pas être mal compris, mais je ne peux m’empêcher de remarquer que le nombre doit toujours être un moyen, jamais une fin. C’est sous n’importe quelle forme, car si techniquement la géométrie est une chose et le nombre en est une autre, ils peuvent être interchangeables dans certaines circonstances. En plus du fait que la plupart des gens ne comprennent même pas quand un auteur commence à dessiner des triangles et des carrés et à établir des relations, «nombre d’or», etc.

La conception, effectivement, des vitraux, est expliquée dans l’article Schémas géométriques des vitrauxet le lecteur peut y accéder directement et éviter la discussion introductive que je fais, car notre culture visait à comprendre la réalité de manière dite «scientifique» , pour lequel les nombres ou les mathématiques sont fondamentaux. Ce n’était pas toujours comme ça et ce n’était certainement pas au Moyen Âge. J’ai essayé de mettre cela en perspective dans deux articles, qui peuvent être lus par toute personne curieuse ou intéressée par cela. L’un des articles est la géométrie et les nombres et l’autre est celui mentionné ci-dessus sur les schémas géométriques. Ces deux articles. avec celui-ci, qui continue à élaborer là-dessus, ils forment le cadre de ce qui serait dans la tête de ceux qui ont conçu et exécuté les vitraux gothiques de ces cathédrales.

Realidade e Números

Sinto que o espírito sente e vê mais do que raciocina e mede. Não há necessidade de um treinamento de anos em matematicas para perceber a leveza, ou a beleza, ou o equilibrio de certas construções. Quem tiver percorrido o texto de Charpentier, com certeza encontrou número em quantidade suficiente para satisfazer qualquer gosto, porém, eu ai coloquei em parte para seguir o autor que eu estava traduzindo (ou introduzindo) e em parte pelo desafio que sempre isto representa.

Mas não valorizo muito isto. Talvez o Universo seja número. Mas não é para mim. Tenho forte convicção que número não existe na cabeça do Criador, apenas na do homem. Os grandes pensadores da antiguidade julgavam que tudo era número porque tinham a ilusão de conseguir apreender completamente o Universo em alguma fórmula. Na verdade, porque a cabeça do homem é limitada, isto é impossível, bastando citar, por exemplo, a teoria da Relatividade, com o tempo variável ou, por exemplo, as contradições da Teoria da Evolução de Darwin, que são dois dos mais finos exemplos do que o cérebro humano foi capaz.

Vou mais longe. Esta limitação ou fraqueza, acaba virando uma vantagem, pois já que efetivamente não dá para entender, vamos entender o que dá e, para isto, nossa cabeça vem muito bem equipada. Seria uma falha elementar tanta assertivação na nossa “imagem e semelhança com Deus” e que o “Universo está a nosso dispor”, se o equipamento básico para se validar isto viesse básicamente com um defeito tão grave de fabricação…

Vale a mesma crítica para as pretensões “científicas”… sem querer valorizar a verdade que aqui encontramos e que, é verdadeira apenas para quem a encontra…como aliás também é o caso de qualquer verdade.

Vou intencionalmente reduzir aqui o papel da Geometria e dos números e para tanto selecionei do texto de Painton Cowen (que coloca bastante geometria, mas não indica prioridade dela) e que compartilha de minha percepção, pois sem ser tão enfático, também afirma que é meio e não fim.

De qualquer forma, discuto em outro local a questão da Geometria e números.

O Design de Chartres

O estilo e o design dos vitrais e rosáceas do século 13 naturalmente tem muito em comum o aspecto que cada santo, anjo, profeta ou virtude tem um área muito pequena para se expressarem e o pano de fundo é o menor possível. A noção, assunto ou objeto focalizado quase sempre está dentro de uma circunferência que por sua vez está dentro de alguma pétala da rosácea, sendo que o espaço restante é preenchido por mosaico decorativo ou ornamentação em forma de folha.

O lay out geométrico é discutico em mais detalhe acima clicando-se Geometria e é um triunfo de design. A maioria das rosáceas se apoia geometricamente no desenho da cruz e do circulo em padrão radial e a roda, predecessora da rosácea, se faz sentir mesmo nos vitrais do fim do período. As rosetas e as combinações de círculo quadrado que são encontradas tão frequentemente na decoração gótica medieval, não naturalmente favoritas nas rosáceas, como podemos ver por exemplo nos vitrais de Lausanne, Clermont-Ferrand, Paris et Seés, etc.

Em Cantembury, o design da rosácea é quase identico ao enorme mosaico atrás do altar inspirado no padrão islamico de quadrados e círculos. A inspiração não é apenas islamica, pois os celtas e até mesmo os romanos em sua cerâmica de piso os usavam.

O desenho em si dos vitrais envolviam um trabalho composto de quatro setores de atividade:

• O Clero
• O pedreiro mestre
• O ferreiro e
• O mestre vidreiro

Este grupo que determinava a escolha e disposição do program iconografico que era executado.

Haviam algumas escolhas, como por exemplo o mestre vidreiro seguia uma tradição bastante estabelecida no sentido de colocar S.Pedro sempre com uma longa barba, S.Paulo careca ou Melchisedec com um cálice de sacerdote nas mãos. Algumas vezes o personagem carrega rolos (como de pergaminho) com seu nome, como na Rosácea Norte de Chartres. A escolha da cor também seguia um padrão tradicional, com o azul de fundo contra o qual o vermelho, verde, marrom e o púrpura eram colocados ocasionalmente com contraste violento. Em alguns caso o vermelho é usado como fundo, isto ocorre quando Cristo ou sua ancestralidade é o assunto principal.

Porém, apesar de muitas regras e convenções existia amplo espaço para a expressão individual. O mestre vidreiro desenhava o laoyout e o detalhe selecionando o vidro pelas suas caracteristicas naturais, cobria as linhas de estanho em volta dos contornos, pintados em detalhe, e após o tratamento com fogo final, ajustava cada peça na estrutura.

É fascinante imaginar que na criação de uma rosácea de 12 metros de diâmetro – como as de Chartres e Paris – não havia como se imaginar como ela iria ficar no seu lugar quando o projeto estivesse completo. Devem ter havido períodos de experimentação, porém é sem duvida uma medida da habilidade extrema, conhecimento e intuição que estes artesãos medievais tinham, pois virtualmente trabalhavam cegos nestas vastas composições. Não importa o quanto o design fosse pensado e desenhado e experimentado, era quase impossível saber antecipadamente como atuariam fatores tais como o efeito da grossura do vidro e o angulo do sol em diferentes horas do dia na sua forma final.

Observação: Penso exatamente o oposto de Painton Cowen. É isto exatamente que dava a medida do talento e da habilidade envolvidos haja visto que existe, no interior da catedral de Chartres, na parte inferior do lado Oeste do transepto Sul, uma pedra retangular, chumbada obliquamente nas outras lajes, num talhe branco distinto sobre o cinza escuro geral do lajeamento. Esta pedra esta marcada por um pino de metal brilhante ligeiramente dourado.A cada ano, em 21 de Junho, assim que o sol nasce, o que geralmente acontece nesta época, um raio de sol vem, exatamente ao meio dia, bater sobre esta pedra branca, um raio que penetra por um espaço preparado no vitral conhecido como de São Apolinario, o primeiro da parede Oeste deste transepto.

Painton Cown completa:

O efeito de halo, em função do qual as cores mais leves tendem a se espalhar em cima das mais pesadas, frequentemente resulta numa mudança substancial na cor total do vitral quando visto à distância. O efeito de halo também influencia o equilibrio geral das cores, sendo o azul a cor mais problemática, pois escurece todo o conjunto dependendo de como é iluminada. Estes efeitos são mais pronunciados quando o sol está contra o vitral – como é o caso em Reims – onde o equilibrio das cores é mudado substancialmente e o vermelho predomina quando o sol está se pondo.

O que conta no final é que uma Rosácea bela é aquela que apresenta uma nova face a cada hora do dia e a cada dia do ano. É como uma garande sinfonia, que nunca cesa de nos fascinar, de uma variedade infinita dentro de uma ordem ciclica e perfeita, de ritmos irregulares dentro de compassos regulares ou compassos irregulares dentro de ritmos regulares, de luz, de sombra e de cor sutilmente combinados ou dramaticamente contrastantes.

É como se nestas grandes obras de luz e geometria, o artista seja um criador de esferas, atingindo a música das esferas do Cosmos.  Sobre isso, leia os seguintes artigos de Marcelo Gleiser publicados na Folha de S.Paulo

Musica das Esferas 2001

Musica das Esferas 2003

Musica das Esferas 2007

 

 

Voltar para plano dos vitrais

Schéma géométrique du vitraux de Chartres

                            

Veja em Português        See it in English

L’intérêt presque obsessionnel des architectes et théologiens médiévaux pour la géométrie remonte au moins à l’époque de Sto Agostinho (354-386 après JC). Il y avait une attente mêlée au mysticisme des nombres en termes pythagoriciens et néoplatoniciens, en tant qu’expression de la proportion divine.
Sto.Agostinho parle de la fonction «anagogique» (d’élever l’esprit en contemplant les choses divines) de la musique et de la géométrie et de la beauté obtenue à travers les figures et les proportions. A Chartres, le souci du Chancelier Thierry pour les nombres se manifeste dans ses représentations géométriques de la Trinité et crée une tendance qui réduit presque la théologie à la géométrie, ce qui dérange un autre Chancelier, John de Salisbury. Cependant, la géométrie est à la base de toute cathédrale gothique, tout étant créé à travers ses relations de base.

Os meios de proporcionamento fazendo a elevação da figura do plano para três dimensões, usando polígonos regulares, especialmente o quadrado, foram cuidadosamente guardados até que no século 15, Matthew Roriczer de Ravensgurg tornou o conhecimento público.

A aplicação da Geometria para a construção e o proporcionamento do predio será discutida em separado nos Les mystères de la cathédrale de Chartres

A Geometria e o uso de polígonos para o estabelecimento das proporções, é fundamental para todos os vitrais de rosáceas. Qualquer uma delas envolve um cuidadoso cálculo e uma construção precisa. Isto opera de maneira diferente para cada rosácea e das satisfações que podem ser obtidas, a maior não é a visual, mas a intelectual, cósmica em suas implicações.

Passamos a analisar quatro das maiores rosáceas existentes na França e no mundo, afim de substanciar toda esta discussão um tanto estéril fim de dar insight tanto para como foram construidas estas rosáceas, como para demonstrar as relações geométricas secundárias, que emergem naturalmente de qualquer rosácea perfeitamente desenhada.

A rosácea do portão Real ou Oeste de Chartres

Acima, figura 1

Esta clássica rosácea acima do Portal Real (Central da entrada), na fachada Oeste, é mais que um triunfo de design: é um “tour de force” onde cinco sistemas separados de sistemas de proporcionamento pulsam ritmicamente através do círculo, dividindo as espirais a partir do vortex.
A frase acima foi tirada do artigo de John James sobre a geometria desta rosácea, uma análise soberba que servirá de base para este nosso trabalho.
Trabalhando a partir das melhores medições disponiveis de desenhos o mais acurados possíveis, James raramente aceitou um erro de +/- 2 cm para a tarefa de estabelecer as relações dentro desta imensa rosácea.
Ele descobriu que foram usados dois sistemas de medidas (“métricos”), um mostrado acima, que é romano com o pé no valor de 29,6 cm (portanto, onde está marcado 10´é igual a 2,96 mts e onde está marcado 22´ é igual a 6,51 mts)
O autor menciona, (e não faz sentido), que John James mediu 3´ e 10´, que reduz para 2/3 e 1/3, dando 6 2/3´e 2 1/3´,que ele menciona ser uma expansão de 3 e um eco elegante da preocupação de Thierry com a configuração da Trindade.
Observo que o autor omitiu ou não entendeu o texto original, pois a observação não faz sentido. Creio que John James mediu o raio de 1 a A como sendo 1/3 de 1 a C como 2/3. Onde ele menciona 3´poderiam ser apenas 22´, raio de 1 até o limite do círculo, uma vez quede 1 até A, B, C, ou D, nunca vai dar isto. Aliás, a medida 22´não é coerente, pois o centro “C” deveria estar a 11´e não a 10´como está. Porém se considerarmos as linhas de centro A, B, C, D e E como relacionadas de 1 até o limite do círculo, menos a borda, a referência fica 20´e tudo encaixa, ficando apenas problemas de precisão à medida estabelecida, não importando o sistema métrico. Pelo jeito, Lassus se confundiu e John James seguiu.
Ótimo exemplo para validar meu ponto. Como eu disse, nem o especialista entende. O problema todo reside que o sistema métrico ainda não existia (e se passássemos para metro, a ressonância elegante com o numero 3 iria para o espaço, indicando que ou Deus não ressoa com isto, ou não é onisciente, pois se ele sabe o futuro deveria antever o sistema métrico…)
Ainda, na figura 1 acima, um segmento da rosácea, de um desenho de Lassus de 1842, mostra que a unidade de 10´dá a linha de centro da fileira de rosetas (“C”) e a medida de 6 2/3 ´ dos medalhões (“B”) internos. As doze pétalas da rosácea central estão no círculo que gira em “A”. Os medalhões que estão centrados em “B” e “C” definem as juntas de alvenaria ao fim de cada pilarete.

Acima, figura 2

Na figura 2, acima, as 12 rosetas maiores (“D”) são definidas com a unidade de 3 pés criando 6 quadriláteros de lado 12´, criando uma estrela de 24 pontas (em”C”) e os pontos de contacto das rosetas.

Acima, figura 3

Pela interconexão do centro das rosetas “B” forma-se três quadrados (retângulos?) definindo a cornija (a moldura no topo – ou na base – da coluneta) e quando os quadrados são expandidos em uma estrela de 12 lados eles encontram o centro das “lunetas” “E”. Esta mesma estrela é também tangencial às rosetas e à fileira do meio dos “medalhões” “C”. Finalmente um hexágono em volta da estrela define o perímetro das folhas esculpidas e, como John James diz: “Com uma precisão elegante o Mestre colocou a folha que adorna o centro de cada capitel na interseção dos braços da estrela. Desta forma, numa consistência esplendida as terceiras fixações prendem tanto as formas das folhas como os três anéis concêntricos dos círculos que parecem um girassol”.

Acima, figura 4

Existe uma construção (geométrica) final da unidade de 3 pés que gera o círculo externo. Em volta do centro de 3 pés quatro triângulos equiláteros são desenhados e os três quadrados que os contém são expandidos em uma estrela de 12 pontas que chega até o perimetro no eixo de pequenas “lunetas” “E”.

John James leva a análise ainda mais profundamente, encontrando uma série de “pentágonos em expansão” da unidade central de 3 pés, que confirmam certos pontos na estrutura, porém Painton Cowen acha isto menos convincente. James encontra ainda muitas outras interrelações sutis e ao se usar uma segunda unidade de medida, o pes manualis, que é 6/5 do pé Romano (3,55 mts) “cada elemento por menor que seja e cada molde reflete de uma forma ou outra as medidas básicas. Na geometria cuidadosa, tudo é feito como parte de tudo o mais, nada fica sozinho. Desta forma foi com o universo de Deus, e desta forma o homem deveria ter como objetivo louvar isto Dele”. (?)

A rosácea no transepto Norte de Chartres (Rose de France)

Acima, figura 5

A Gloriosa Rose de France no transepto Norte de Chartres foi usado no livro de Painton Cowen para ilustram muitos e diferentes aspectos das rosáceas. Sua geometria divina está entre suas glorias mais gloriosas. Tudo nesta rosácea é gerado das propriedades do quadrado dentro do círculo (“quadratura do círculo”?….)

Os 12 quadrados conjugam angulos em relação aos raios e são as caracteristicas desta rosácea que mais impressionam. Eles podem facilmente ser relacionados com os três quadrados que se interpenetram e contem outros quadrados menores (o olho humano pode ver isto sem auxilio de nada). Estes quadrados sugerem que existe uma sutil geometria subjacente que pé baseada na espiral e o olho parece sentir que a forma de girassol que expande em espiral para fora do centro.

Na verdade, o conjunto de quadrados, que que cria uma verdadeira geometria espiral, está mais relacionado com o trabalho em pedra que fica para a face externa. São os centros das “lunetas” na parte mais para fora do círculo que criam uma geometria de quadrados que se expandem e se encolhem formando as janelas quadradas. Pela junção de todo segundo e terceiro quadrado a sequencia pode ser desenhada e eles muito elegantemente para a roseta central abaixo, com suas doze pétalas.

Esta série de quadrados pode também ser relacionada com o Número Áureo Phi, que já vimos por aqui.

Vamos abrir um parêntesis que o Cowen não explicou direito esta maravilha.

O número áureo, 1:1.618034, representado pela letra grega phi, é um número que ocorre naturalmente, como o pi (3,1416…) que repetidamente ocorre em várias relações. Como o pi é um número irracional. Diferentemente de pi, ele aparece no padrão de desenvolvimento de muitas coisas vivas (v.Fibonacci) como a espiral formada pela concha marinha, e muitos outras. É na realidade um padrão de crescimento que, se continuado, não falha. Foi derivado pelos antigos gregos e foi usado no antigo Egito no desenho e na construção de pirâmides, prédios e monumentos. Eles descobriram que podam criar um sentimento de ordem natural, bem como de integridade estrutural nos seus trabalhos usando esta relação. Nos anos 30 o Instituto Pratt fez um estudo utilizando várias proporções retangulares em conjuntos verticais e os apresentou a centenas de alunos para que opinassem quais os agradavam mais. Os que estavam baseado no número áureo foram escolhidos consistentemente, o que parece demonstrar que as dimensões nestas porporções geram uma resposta de satisfação.

Fibonacci publicou seus trabalhos em 1202 – cerca de 30 anos antes destas rosáceas serem construidas e de acordo com o autor H.E.Hurley, era amplamente divulgado e provavelmente teve a atenção da Escola de Chartres.

Nesta rosácea de Chartes existem doze grupos de espirais seguindo esta “série de Fibonacci”, como se pode ver na figura 5 acima.

Neste esquema as “lunetas” dos profetas no círculo externo se ligam em pontos chave na estrutura, passando por 12 reis, anjos, pombas, que dão nascimento ao Logos no centro. O Logos criativo do Universo, a lei da natureza, é seguida pelo homem para oferecer perfeita beleza.

Existem outras relações estruturais nesta rosácea, independentemente deste arranjo em série Fibonacci.

As primeiras duas podem ser vistas na figura 6 abaixo, que demonstra a rede de triangulos equiláteros construidos dos semicirculos em torno do limite. Eles criam três quadrados intercalados em torno do centro, cujos cantos definem o centro dos 12 circulos mais internos. Outro sistema de quadrados interliga as 12 pequenas aberturas quadradas com as 12 quadrifolhas, mostrado pelas linhas mais fracas na figura 6.

Na figura 7 abaixo, outro sistema de quadrados interconecta os semi circulos externos às diagonais dos 12 pequenos quadrados – outra demonstração da natureza sutil desta notavel rosacea. Provavelmente as unidades de medidas e as prórpias medidas, se forem estudadas, revelarão outras relações sutis e engenhosas.

A rosácea no transepto Sul de Chartres

Figura 8 acima

A figura 8 mostra um esquema para a rosácea sul de Chartres. Sem dúvida existem outros esquemas mais sutis que este, mas seste mostra interessantes relações que o olho provavelmente sente inconscientemente. O semicirculo em volta da borda sugere um mecanismo de focalização que leva o olho ao centro: as linhas finas no diagrama puxam das extremidades de cada um dos semicirculos do limite externo para um ponto no centro do semicirculo oposto definindo perfeitamente o medalhão central que contém Cristo (e cada um toma os pequenos quadrifoglios no percurso, desta forma definindo seus centros). Qualquer linha desenhada do centro de um semicirculo para o centro do outro, a uma distância de 5 lunetas é tangente não menos que quatro dos 12 medalhões centrais, o que é um alinhamento notavel. Além disto, um conjunto de três quadrados na rosácea é tangencial à fileira central das lunetas maiores. O mesmo sistema de três quadrados pode ter gerado mais relações sutis das propriedades dos três quadrados, porém isto demanda um estudo mais aprofundado.

A rosácea no transepto Norte na Notre Dame de Paris

É oportuno mencionar, pois esclarece a questão.

Figura 9 acima

A Rosácea do transepto norte da Notre Dame de Paris, algumas vezes é chamada de Rosácea dos Alquimistas, provavelmente e o exemplo supremo de uma construção entrelaçada de geometria escondida. A figura 9 mostra como a Geometria cria e define cada camada de progressão lógica.

O ponto inicial é o centro de qualquer uma das 32 luntas externas. Ligando-se umas às outras, como no desenho, a cada 8 lunetas, e repetindo-se o processo, uma linha interminavel é gerada que liga todas estas lunetas externas. Ao mesmo tempo em que cria tangentes a um circulo que contem os centros da proxima camada de 32 medalhões. Repetindo-se a operação, porém em vez de a cada 8, fazer a cada 11, criamos duas estrelas entrelaçadas de 16 pontas e elas, por sua vez, definem os centros de cada uma das 16 lunetas centrais contendo os profetas. Dests centros quatro quadrados entrelaçados podem ser extraidos, que por sua vez tangencialmente produzem o vermelho e o dourado em torno da Virgem e da Criança no centro. Finalmente, uma estrela de 16 pontas deste circulo cria uma abertura para as lunetas centrais – e o tamanho das aberturas que formam o limite da roseta central.

Desta forma a geometria relaciona todas as partes a todas as partes e a totalidade dos pontos de foco – o centro.

Esta Rosacea está plena de outras relações sutis que podem ser verificadas observando-se a figura 9 acima, usando-se uma régua e um esquema similar aos dois descritos, por exemplo, a cada 7, ou 10, etc.

Geometric Scheme of Chartres Stained Glasses

                               

Veja em Português        Voir en français

The almost obsessive interest that medieval architects and theologians showed in geometry dates back at least to the time of Sto Agostinho (354-386 AD). There was an expectation mixed with the mysticism of numbers in Pythagorean and Neoplatonic terms, as an expression of divine proportion.
Sto.Agostinho speaks of the “anagogic” function (of elevating the spirit by contemplating divine things) of music and geometry and the beauty achieved through figures and proportions. In Chartres, Chanceler Thierry‘s concern for numbers was manifested in his geometric representations of the Trinity and created a trend that almost reduced theology to geometry, which disturbed another Chancellor, John de Salisbury. However, geometry is at the base of any Gothic Cathedral, everything being created through its basic relations.

The means of proportioning the elevation of the figure from the plane to three dimensions, using regular polygons, especially the square, were carefully guarded until in the 15th century, Matthew Roriczer de Ravensgurg made public knowledge.
The application of Geometry for the construction and proportion of the building will be discussed separately in Les mystères de la cathédrale de Chartres
Geometry and the use of polygons for the establishment of proportions is essential for all stained glass windows. Either involves careful calculation and precise construction. This works differently for each rosette and of the satisfactions that can be obtained, the biggest one is not the visual, but the intellectual, cosmic in its implications.
We proceed to analyze four of the largest rosettes existing in France and in the world, in order to substantiate this somewhat sterile discussion in order to give insight both to how these rosettes were built, and to demonstrate the secondary geometric relationships that naturally emerge from any rosette perfectly drawn.

The rose window of the Royal or West gate of Chartres

Above, figure 1

This classic rose window above the Royal Portal (Central entrance), on the West façade, is more than a design triumph: it is a “tour de force” where five separate systems of proportioning systems pulsate rhythmically through the circle, dividing the spirals a from the vortex.
The above sentence was taken from John James article on the geometry of this rosette, a superb analysis that will serve as the basis for our work.
Working from the best available measurements of drawings as accurate as possible, James rarely accepted an error of +/- 2 cm for the task of establishing relationships within this immense rosette.
He found that two measurement systems (“metric”) were used, one shown above, which is Roman with a foot of 29.6 cm (so where 10′  is equal to 2.96 mts and where is marked 22´ is equal to 6.51 mts)
The author mentions (and it does not make sense), that John James measured 3´ and 10´, which reduces to 2/3 and 1/3, giving 6 2/3´ and 2 1/3´, which he mentions to be a expansion of 3 and an elegant echo of Thierry’s concern for the Trinity configuration.
I note that the author omitted or did not understand the original text, as the observation does not make sense. I think John James measured the radius from 1 to A to be 1/3 of 1 to C to 2/3. Where he mentions 3″ it could be just 22′, radius of 1 to the limit of the circle, since 1 falls to A, B, C, or D, this will never happen. In fact, measure 22′ is not coherent, since center “C” should be 11″ and not 10″ as it is. However if we consider the center lines A, B, C, D and E as related from 1 to the limit of the circle, minus the border, the reference is 20´ and everything fits together, leaving only precision problems to the established measure, no matter the metric system. Apparently, Lassus got confused and John James followed.
Great example to validate my point. As I said, not even the expert understands. The whole problem is that the metric system did not yet exist (and if we moved to meter, the elegant resonance with the number 3 would go into space, indicating that either God does not resonate with this, or is not omniscient, because if He knows the future He should foresee the metric system…)
Still, in figure 1 above, a segment of the rosette, from a drawing by Lassus from 1842, shows that the 10´ unit gives the center line of the rosette row (“C”) and the measure of 6 2/3´ internal medallions (“B”). The twelve petals of the central rosette are in the circle that turns in “A”. The medallions that are centered on “B” and “C” define the masonry joints at the end of each pillar.

Above, figure 2

In figure 2, above, the 12 largest rosettes (“D”) are defined with the 3-foot unit creating 6 12´ side quadrangles, creating a 24-pointed star (in ”C”) and the contact points of the rosettes .

Above, figure 3

Through the interconnection of the center of the “B” rosettes, three squares (rectangles?) Are formed defining the cornice (the frame at the top – or at the bottom – of the column) and when the squares are expanded in a 12-sided star they find the center of the “E” telescopes. This same star is also tangential to the rosettes and the middle row of the “C” medallions. Finally, a hexagon around the star defines the perimeter of the sculpted leaves and, as John James says: “With elegant precision the Master placed the leaf that adorns the center of each capital at the intersection of the star’s arms. In this way, in a splendid consistency, the third fixations hold both the shapes of the leaves and the three concentric rings of the circles that look like a sunflower ”

Above, figure 4

There is a final (geometric) construction of the 3-foot unit that generates the outer circle. Around the center of 3 feet, four equilateral triangles are drawn and the three squares that contain them are expanded into a 12-pointed star that reaches up to the perimeter on the axis of small “telescopes” “E”.
John James takes the analysis even more deeply, finding a series of “expanding pentagons” from the 3-foot central unit, which confirm certain points in the structure, but Painton Cowen finds this less convincing. James still finds many other subtle interrelationships and when using a second unit of measurement, the pes manualis, which is 6/5 of the Roman foot (3.55 mts) “each element however small and each mold reflects in a different way or another the basic measures. In careful geometry, everything is done as part of everything else, nothing is left alone. In this way it was with the universe of God, and in this way man should aim to praise this from Him ”. (?)

The rose window in the North Chartres transept (Rose de France)

Above, figure 5

The Glorious Rose de France in the north transept of Chartres was used in Painton Cowen’s book to illustrate many different aspects of rosettes. Its divine geometry is among its most glorious glory. Everything in this rosette is generated from the properties of the square within the circle (“square of the circle”?….)
The 12 squares combine angles in relation to the rays and are the characteristics of this rosette that most impress. They can be easily related to the three squares that interpenetrate and contain other smaller squares (the human eye can see this without assistance). These squares suggest that there is a subtle underlying geometry that is based on the spiral and the eye seems to sense the sunflower shape that expands in a spiral out of the center.
In fact, the set of squares, which creates a true spiral geometry, is more related to the stone work on the outside. It is the centers of the “telescopes” on the outermost part of the circle that create a geometry of squares that expand and shrink to form square windows. By joining every second and third square the sequence can be drawn and they are very elegantly drawn to the central rosette below, with its twelve petals.
This series of squares can also be related to the Golden Phi Number, which we have already seen here.

Let’s open a parenthesis that Cowen did not explain this wonder well.

The golden number, 1: 1.618034, represented by the Greek letter phi, is a naturally occurring number, like pi (3.1416 …) that repeatedly occurs in various relationships. Since pi is an irrational number. Unlike pi, it appears in the pattern of development of many living things (v. Fibonacci)) such as the spiral formed by the seashell, and many others. It is actually a pattern of growth that, if continued, will not fail. It was derived by the ancient Greeks and was used in ancient Egypt in the design and construction of pyramids, buildings and monuments. They found that they can create a feeling of natural order, as well as structural integrity in their work using this relationship. In the 1930s, the Pratt Institute made a study using various rectangular proportions in vertical sets and introduced them to hundreds of students to give their opinions on which ones they liked best. Those based on the golden number were chosen consistently, which seems to demonstrate that the dimensions in these portions generate a satisfying response.
Fibonacci published his work in 1202 – about 30 years before these rosettes were built and according to author H.E.Hurley, it was widely publicized and probably had the attention of the Chartres School.
In this Chartes rosette there are twelve groups of spirals following this “Fibonacci series”, as can be seen in figure 5 above.
In this scheme, the “telescopes” of the prophets in the outer circle connect at key points in the structure, passing through 12 kings, angels, doves, which give birth to the Logos in the center. The creative Logos of the Universe, the law of nature, is followed by man to offer perfect beauty.
There are other structural relationships in this rosette, regardless of this arrangement in Fibonacci series.
The first two can be seen in figure 6 below, which shows the network of equilateral triangles built from the semicircles around the limit. They create three interspersed squares around the center, whose corners define the center of the 12 innermost circles. Another system of squares interconnects the 12 small square openings with the 12 quadrifects, shown by the weakest lines in figure 6.

In figure 7 below, another system of squares interconnects the external semi-circles to the diagonals of the 12 small squares – another demonstration of the subtle nature of this remarkable rosacea. Probably the units of measure and the measures themselves, if studied, will reveal other subtle and ingenious relationships.

The rosacea in the  South Transept of Chartres

Transept

Figure 8 above

Figure 8 shows a schematic for the southern Chartres rosette. No doubt there are other schemes more subtle than this one, but this one shows interesting relationships that the eye probably feels unconsciously. The semicircle around the edge suggests a focusing mechanism that takes the eye to the center: the thin lines in the diagram pull from the ends of each semicircle from the outer limit to a point in the center of the opposite semicircle, perfectly defining the central medallion containing Christ (and each one takes the small quadrifoglios in the course, thus defining their centers). Any line drawn from the center of a semicircle to the center of the other, at a distance of 5 bezel is tangent to no less than four of the 12 central medallions, which is a remarkable alignment. In addition, a set of three squares in the rose window is tangential to the central row of the larger bezels. The same system of three squares may have generated more subtle relationships of the properties of the three squares, but this requires further study.

The rose window in the North transept at Notre Dame de Paris

It is worth mentioning, as it clarifies the issue.

Figure 9 above

The Rosette of the northern transept of Notre Dame de Paris, is sometimes called the Rosette of the Alchemists, probably the supreme example of an intertwined construction of hidden geometry. Figure 9 shows how Geometry creates and defines each logical progression layer.
The starting point is the center of any of the 32 outer joints. Connecting to each other, as in the drawing, every 8 telescopes, and repeating the process, an endless line is generated that connects all these external telescopes. At the same time that it creates tangents to a circle that contains the centers of the next layer of 32 medallions. Repeating the operation, however instead of every 8, doing every 11, we created two intertwined 16-pointed stars and they, in turn, define the centers of each of the 16 central telescopes containing the prophets. From these centers four interlaced squares can be extracted, which in turn tangentially produce red and gold around the Virgin and Child in the center. Finally, a 16-pointed star in this circle creates an opening for the central bezels – and the size of the openings that form the border of the central rosette.
In this way, geometry links all parts to all parts and the totality of the focus points – the center.
This Rosacea is full of other subtle relationships that can be verified by looking at figure 9 above, using a ruler and a scheme similar to the two described, for example, every 7, or 10, etc.